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Handbook of graph theory
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN HANDBOOK OF GRAPH THEORY EDITED BY JONATHAN
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DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN
HANDBOOK OF
GRAPH THEORY EDITED BY
JONATHAN L. GROSS JAY YELLEN
CRC PR E S S Boca Raton London New York Washington, D.C.
DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor
Kenneth H. Rosen, Ph.D. AT&T Laboratories Middletown, New Jersey
Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs Charalambos A. Charalambides, Enumerative Combinatorics Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications with Maple Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Crytography Richard A. Mollin, Quadratics
Continued Titles Richard A. Mollin, RSA and Public-Key Cryptography Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Douglas R. Stinson, Cryptography: Theory and Practice, Second Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography
8522 disclaimer.fm Page 1 Tuesday, November 4, 2003 12:31 PM
Library of Congress Cataloging-in-Publication Data Handbook of graph theory / editors-in-chief, Jonathan L. Gross, Jay Yellen. p. cm. — (Discrete mathematics and its applications) Includes bibliographical references and index. ISBN 1-58488-090-2 (alk. paper) 1. Graph theory—Handbooks, manuals, etc. I. Gross, Jonathan L. II. Yellen, Jay. QA166.H36 2003 511'.5—dc22
2003065270
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microÞlming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of speciÞc clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-090-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. SpeciÞc permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiÞcation and explanation, without intent to infringe.
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PREFACE
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History of Graph Theory
HISTORY OF GRAPH THEORY
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4.2
Chapter 4
CONNECTIVITY and TRAVERSABILITY
EULERIAN GRAPHS 8 ., A 6 A 6 A ; 2 2 4 L 6 A 6 . 7 6 A 6
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215
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217
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219
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221
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CONNECTIVITY and TRAVERSABILITY
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223
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CONNECTIVITY and TRAVERSABILITY
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231
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CONNECTIVITY and TRAVERSABILITY
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CONNECTIVITY and TRAVERSABILITY
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A F 2 8 5 % %8 =/4> 1K9 -A 970 : A =97> 9 3K 1 ? ; " L 9 K1 -( 330 ( .# + 33 49K7
-(310 * ( A : J I & . I J + - ) =/31> 4K79 -(3 0 * (
. H 8 N $ =A> /3 K 7 -(370 * ( 5 +
- ) =/37> /K1 -(330 * ( ! ' =/33> 3K 3
-(3/0 * ( A = > - $ % /39
$ =/3/> F K 4K/ -(/10 * ( * ) $ F * //1
/
-(/0 * ( * ) $ F * //
/ = .
.
8 1 * ) )57(' 1 %
$ . =A> J!
-(110 * (
111 /K39
-(10 * ( = > 6 = > $ =11> K -(10 * ( 8 % 5 $ =11> 99K3 -((/10 * ( ( & ; + - =//1> 4K4
Section 4.2
Eulerian Graphs
235
-(2340 * ( $ 2 ; ! =/34> 7K79 -(+3/0 * ( A + $ , # =/3/> 44K71
%
-((70 : ( . (& -. '. N FH /7 I -* 390 * N $I & : + N $ 12 =39> 1K -J470 J A ? ! B =
&> $ -> %8 =/47> F K7 ? -J730 J $ ! $ -> =/73> F 97K31 G : )4 4 $4 (/ 2 KL , -: 3/ 0 $ A 3/ -$3 0 6 $J ' " ! $ =/3 > 9K -F 9/0 H F + H + =A> ; N U $ J /93 ) ' $ ( =/9/> 39K/9 -F 34 0 H F + . A % 8 " =A> /34 $ % 0 ' % N : =/34> 9K4 -F 34 0 H F + A + 0 $ % =/34> F 9K/ - 3/0 H . 6 I 2 $ =3/> /K1 - 990 2 6 ! & & $ 5 ) @@/ * =A> ( ; N * K /97 -9/0 . ) H 8 N $ =A> F! 9K3 -3 0 . F! 7 )! + - =/3> 1K4 -9/0 * & =/9/> 19K13
236
Chapter 4
CONNECTIVITY and TRAVERSABILITY
-&30 . J & A
- % J $ J * < =A> $ /3 : F $ 8 F! /7 -270U
C D H & A =-A74 0>
4.3.1 The Basic Problem and Its Variations DEFINITIONS
'
! &
'
2 , ? = > ! ! ' & & & !
.
6
' 6
' 6
&& - && && ! &&
' 6 && $ && *,' FACTS
' '
N .
= > $ F = >
238
Chapter 4
CONNECTIVITY and TRAVERSABILITY
The Eulerian Case DEFINITIONS
'
= > ! & , # ! &
= >
#
'
#
'
! &
'
"
? =
B
" #
!
> #
( 3 3 (
& =
*! # C D
>
=
>
( 3
FACTS
'
#
'
'
"
= #>
REMARK
'
# $ #
Variations of CPP DEFINITIONS
'
'
6 B
= >
'
. '
,
; '
'
. '
, B
= >
'
'
" N
6 !
# CD = !
>
" ! ! @ !
'
'
&' * *
6
! B
Section 4.3
239
Chinese Postman Problems
6 ! ! ! = > # !
'
'
6 #
FACTS
'
6
"
½
¾
, = > B , #
!
4
!
4
½
¾
! = , > !
Æ
, ! !
"
U
* >
'
½
! ,
¾
&"
=
(
! , B
=
'
-AH90>
-+ /0 6 ! F
'
6 F
M ! = -2 H9/0>
REMARK
'
" # #
!
# -A 2: /4 0 -A 2: /4 0 A#
-(/0
4.3.2 Undirected Postman Problems 6 2 ! # = >
6
) ! ! A =-A74 0> !
DEFINITIONS
'
$
# =$
'
! !
>
#
240
Chapter 4
CONNECTIVITY and TRAVERSABILITY
Algorithm 4.3.1: , $ 5677 #' 5' $ !
! !
: ! ( ( 3 ( ! ( 3 ( # ! ! ( ! 4 ( 4 . :
6
! ! @ ! !& A = -A74 0 -A74 0> 6 , 6 ! ( = -J 790> ' 2 ! & #
! =( %
> EXAMPLE
' ( U ! , 6 ! !' L#
: 4
Figure 4.3.1 $ $! " '
! ( ! 4 6
Section 4.3
241
Chinese Postman Problems
½ ½
6 ! ( 6
= ! 1> ! & !'
REMARKS
' 6 !
# ! ! ! & = & > ! = -2 H9/0>
' " ( B
B # ! A H =-AH90>
' 6 # ! & N ! * , ! ! , ' !
! ! &
" !
8 ! ! ! = > ! ! !
4.3.3 Directed Postman Problems 6 . " !
6
FACTS
' ! &
' 6
! U
6 & " /& ; = >
242
Chapter 4
Algorithm 4.3.2:
CONNECTIVITY and TRAVERSABILITY
, $ 677
#' ! !
5' $ ! " A
( ? 5= > )&5= > ! '
¾
(
(
¾
( 1
¾
( ?
( ( = >
Producing an Eulerian Tour in a Symmetric (Multi)Digraph
B = > 6 ! = -A840> DEFINITION
' ! # Algorithm 4.3.3:
7" ! " $" !"
#' A 5' A
# £ ( £ ( # ? £ : ! 5 = > : £ # £ ! !
!
? £ ' # = > = > # U
!
Section 4.3
Chinese Postman Problems
243
EXAMPLE
' ( , #
#
# , ! ' ( ? ( ? U ( ? U ( ? 1 ! ! , = > # ! ( 6 ! Æ# ! # , = > ! # B'
Figure 4.3.2 $ $! " ' REMARKS
' 6 !& )! A H =-AH90>
' 6 B ' " B ! # # ( # (
!
4.3.4 Mixed Postman Problems FACTS
' 6 # $ - U 6"("8":"6< = - 970>
244
Chapter 4
CONNECTIVITY and TRAVERSABILITY
' $ - ! # = > # ! ! = -2 H9/0
Deciding if a Mixed Graph Is Eulerian DEFINITIONS
' 6 #
' '
#
# # # B
#
' # , + = > @ ! + = > + = > + + 5 +
= > + = -((70> FACTS
'
= > # ,
'
$ #
EXAMPLE
'
# ! # 6 ( , # B ½
Figure 4.3.3
$" 8 9 +"
6 = > B #
, # ! 6 ! & ! , 6 !& )! !
Section 4.3
245
Chinese Postman Problems
! 8 9 +" 2 $"
Algorithm 4.3.4:
#' # 5'
( ? 5= > )&5= > : ! ! !& )! '
¾
(
(
1 (
( ?
= >
" = > ( " ( ? ; A ( ? ; A : A = >
EXAMPLE
'
6 = #> ( ( " ,
! # # 6
( ( , = > = > ! , 6 # !
Figure 4.3.4
$ $! " ' !" !"
246
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARKS
'
6 ., & & = @ ! > 5
'
" !
= > ! =! @> # ! = -AH90>
The Postman Problem for Mixed Graphs
$ F ! + & ! !
,* * ! ! , ; # *! ! B @ # ! = -A 2: /4 0 -A 2: /4 0> DEFINITION
' , REMARK
' " $ !
# " = ( 4> ! ;! ! M FACT
'
#
6 # = > 6 , # ! ( & =-(9/0>
Section 4.3
247
Chinese Postman Problems
EXAMPLE
' 6 B # # ( 4 U ! " # ! F! ! ! , U ! B ; ! ' U N !
Figure 4.3.5 2" : ''" /!"
" ! ! !
!& )! U
-AH90 " # @ ! ! M
Approximation Algorithm ES
6 ! # ! 5 M 6 ! = -AH90 -(9/0> 4 C M D Algorithm 4.3.5: " 9 ' $! " ' ,
#' #
5'
! M !
N # , : ; M :
6 # 4 /#
248
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARK
' = > & A " ! ! ( & FACT
' -(9/0 6
A
# "
A# 7 ! EXAMPLE
' # ( 7 ! ! , 6 A = > ; ! ! E 6 6 ! * ! ! 6 , ! " ! E 16
Figure 4.3.6 $ " 9 ' $! " ' ,
Approximate Algorithm SE
# A ! =#/> = -(9/0>
6
7 = -(9/0> M 4
Section 4.3
249
Chinese Postman Problems
Algorithm 4.3.6: " 9 ' $! " ' ,
#' #
5'
! M !
# # : N : = >
EXAMPLE
' ( 9 A *!
Figure 4.3.7 $ " 9 ' $! " ' , Some Performance Bounds FACT
' -(9/0 6 A
! # A# 9
! EXAMPLE
' ( 3
A ! A ! ! A
Figure 4.3.8 $! " ' , , "& : "
250
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARKS
' " ! # 4 =A>
7 =A> *! A# 3 ' # ! A = > ! A ! ! ! A
' 6 A# 3
B !
!' + A A ! @ T "
C D ! + !& @ B @
6 , ( & =-(9/0> ! ! ! ! ! !
# ' " =-(9/0> ( & 6 ! !
' +
-(9/0 ! U ! ! ( / ! 8 /// ! , ( & # !
L = L//0> ! 6 ( /
Figure 4.3.9 : " " ' $! " ' , ,
' (! $ &
*! !
F #
! B &
6 $
" , '
$ B @ = -8 6/0 -8 6/0 1 > 6 - - &'-
Section 4.3
Chinese Postman Problems
251
References -889 0 A : 8 : . 8 F!& L $ + '. =/9 > 74K/ -8 6/0 8 8 & 6 . . 2 %($ + $ =//> 3K41 -8 6/0 8 8 & 6 2 : 6 . 2 ( 9 =//> 444K43 -8310 8& 6 $ # F!& ( 9 -A74 0 H A $ # $ ! 1 L + ) ' , % 7/8 =/74> 4K1 -A74 0 H A 6 (! + $ 9 =/74> /K 79 -AH90 H A A H $ A 6 $ 4 =/9> 33K -A840 6
A F 2 8 6 ; : 2 % %8 3 =/4> 1K9 -A 2: /4 0 A $ 2 2 : "' 6 5 ) =//4> K -A 2: /4 0 A $ 2 2 : ""' 6 5 ) =//4> //K -(/0 * ( A 2 6 L $ 41 F * =//> -((70 : ( . (& -. '. N FH =/7>
252
Chapter 4
CONNECTIVITY and TRAVERSABILITY
-(9/0 2 ( & # +
$ 7 =/9/> 43K44 -2 H9/0 $ 2 . H ( 1 ' +* ( F! -270 $ 2 2 N A ; $ =/7> 9K99 -23 0 $ 2 + $ ) *6 =/3 > K/ = > -23 0 $ 2 ; +
$ / =/3 > K 7 -J J9/0 J 2 J 6 $ #
$ =/9/> 3/K1 -J 790 J - F! -: Q330 7 K7 3 -F /70
- 970 * ; # A 6 + $ =/97> 4 K44 - / 0 + :
& $ # F!& 5 ) 0 7 =// > K - :/40 + : $ : $ # F!& 5 ) =//4> 9/K 3/ - L//0 8
H L ¾¿ # $ # %($ + $ =///> 4K - /0 6 J ; $ # 5 ) 0 =//> K9 -+ 3/0 Q + ; + A 2 $ =/3/> /9K
Section 4.4
4.4
253
DeBruijn Graphs and Sequences
DEBRUIJN GRAPHS AND SEQUENCES $ %&' ( ' .8 5 2 8 2 8 5 B F 2
Introduction F 8 5 , = > ( ! 8 5 " ! 8 5 ! 8 5 B
4.4.1 DeBruijn Graph Basics DeBruijn Sequences DEFINITIONS
'
5 , & #
? !
6! 8 5 B C BD
' " 7 / $ & &¼ & 6 ! ¼ 7 & , 7
' " 7 / $ B ! ! 7
/
' ? ½ ¾ ¿ ? ·½ ? 6
' '
½ ¾ ½ ¾
½ ¾
½ ¾
? ½
?
½
254
Chapter 4
CONNECTIVITY and TRAVERSABILITY
FACTS
'
! 8 5 B @ B
' 6 8 5 &
& 8 5 B
EXAMPLES
'
1111 8 5 B " 111 11 11 1 1 1 11
'
11111111 8 5 B
DeBruijn Graphs
8 5 B
! B DEFINITIONS
'
=> !
! B L# 5 #
8 5 =>
A ,
# ! ! # !
' 6 8 5
' 6 8 5 8 5 EXAMPLE
'
( ! 8 5
FACTS
' 6
8 5
' A # 8 5 6 , ! 1 ,
' '
A # 8 5 A 8 5
Section 4.4
255
DeBruijn Graphs and Sequences 0000
000 0001
1000 1001
001 0010
100 0100
010 0011
0101
1100
1010 101
1011 011
1101
0110
0111
110 1110
111
1111
Figure 4.4.1 -" . !" ""
' A 8 5
' 6 = > 8 5 => ! 8 5 B 6 B , # ' -" .& "' -8 90 ( 8 5 B
½
¾
4 7 7 1 3 791337
¾
½
REMARKS
' 8 5
8 5 *! 8 5% ! 8 5 B
' 8 5 8 5 => ! 8 5 ! #
4.4.2 Generating deBruijn Sequences Æ 8 5 B , 8 5 & A 8 5 #
256
Chapter 4
CONNECTIVITY and TRAVERSABILITY
FACTS
' -2 70' ! # A ' " 8 5 => B #
# =6 B , , 8 5 > ' 6 B A 8 5
8 5 B E
EXAMPLE
' ( 8 5 B 8 5
0000
000 0001
1000 1001
001 0010
100 0100
010 0011
0101 1011
011
1100
1010 101
0110
0111
1101 110 1110
111
1111
Figure 4.4.2 $" "
=>
REMARKS
' 6 ( ( Æ A =>
# ! ! ( ( ! B 8 ( ( ! W1%
W% ALGORITHM
' 6 8 5 B ( % =B > A 8 5 = > 6 B A =( % >
Section 4.4
257
DeBruijn Graphs and Sequences
Necklaces and Lyndon Words
( & J -(J990 & 8 5 B DEFINITIONS
' = B
>
' B
' 0 & ! : & + & #
B
& FACTS
'
& : ! ! !
' 8 8 & 8= >
! 8= >
- 0
'
-(J990' " =# > : ! ! # ! 1 8 5 B # REMARK
' 6 - = > & ! # ! + - =1> ? 13 - =4> ? /
-
=4> ? 3 !
EXAMPLES
'
( , B
4 01101 11010
0
1
1
10101 01011
0
1
10110
Figure 4.4.3
'
0$ ""
6 : ! 1 + 11 8 5 B
258
Chapter 4
CONNECTIVITY and TRAVERSABILITY
'
6 : ! 11 1 + 1111 8 5 B
'
+ ! ( (4 ? : !
" #
111 11 1 1 " ! ! ! ! # 8 5 B 11111111
4.4.3 Pseudorandom Numbers ( $ = >
@ 8 $ B .8 5 B ! ! DEFINITIONS
' B B
' -2790 6 1 B
!'
6 %
@ 1%
B 6 ! 5 = E > ! ! = E 11 11>
' B ! ' 1 2 8 5 B # 1% # %
'
-390 6 .
+
,
= >
7 / /
! 7=/> 6 , / B +
Section 4.4
259
DeBruijn Graphs and Sequences
FACTS
'
; 8 5 B
B
'
8 5 B , 2 % , ! (
% # B 1% !
B #
'
" = # > 8 5 B
&
;
B 2 5 8 5 B ! ! B
'
-H/0' 6 !
B
8 5 B
4.4.4 A Genetics Application 6 .F # Æ 8 # # !& 6
.F ! 6 #
, B # ! .F B B #
6 Æ 6
+ - 6 + 10 , 8 5 !
B 7
B
DEFINITIONS
!)
'
'
( + ?
, B 2 6
½
7 7¾ 7
.F B ! , +
+
6!
5 .F B 7 ! ,
REMARK
'
A B & +
8 5 Æ !
6 = > B
! # ! @ + 8 5
260
Chapter 4
CONNECTIVITY and TRAVERSABILITY
References -;0 " & & : ! 8 5 B
!!! M M M&MF& M -390 2 H ( N /39 -. 70 F 2 8 5 ' 943K97
/
-(J990 * ( & " J :# 8 5 B + % =/99> 9K1 -H10 H H ; B /7K1 8 * B!A = : 8 . > L 11 -2 H9/0 $ 2 . H ( 1 ' + * ( X /9/ -2790 + 2 % ) % * . /79 -2 70 " H 2 F + 0 $ % =/ 7> 79K 9 -22$ ://0 H ( 2 Æ + $ 2 H * $ :! $ + * ( /// -2
Section 4.5
4.5
261
Hamiltonian Graphs
HAMILTONIAN GRAPHS '
' )*
4 * 4 6 & 4 A# 4 $ 6 ; * T 44 2 47 (
4.5.1 History F = -2 H9/0> ! F + ! * 341 * # # *
. 349 6 ,
6 ! & ! 34/ ! , ! * % 4 * , B " -J 470 344 6 J & B ' 2 ! , = > # 6 J & & B * N J & ! ( -8 :+ 370 DEFINITIONS
' ' '
= >
4.5.2 The Classic Attacks 6 @ 6 & Æ U #
262
Chapter 4
CONNECTIVITY and TRAVERSABILITY
6 # ( Degrees
6 # Æ = >
Æ =
>
DEFINITIONS
' + = > ! 6 = >
' 6 ! = > 5 5 !
' ( ' ? = 9 > = ? 9 > 5 5 ( 3 9 ! E
6 ! '
: =
>?
/ (
(½ (
FACTS
' '
-. 40 " :
-;710 " : = >
Æ = >
: = >
-;70 " := > E
EXAMPLE
'
! ! # , = ( 4> 6 * Æ = > ? =* > := > ? * . % 6 ;% 6 =( > 6 ! Æ = > ? * : = > ? * ! ; = ( 4>
Figure 4.5.1
' -H 310 :
2$$" ! " " & ;"& "$
!
Section 4.5
263
Hamiltonian Graphs
' -$$70 " ? = 9 > = > ! => E => E 5 9
'
-8970 :
! =
6
> ! ,
!½=
!=
>
>
' -*/0 REMARK
'
6 # 6 = > & , *!
! " #
1 6 ! ! ! =! >
Other Counts DEFINITION
' 6 # ( - =(>
5 ( - =+ > + 5 # +
! "
+ ! ,
' 6 = > !
#
'
= >
EXAMPLE
' 6 =5 *> ! * # + ! + ? ? 5 ? * 5 ! ! 5 + * + Y ¾ ( 4 ! ! = 7> = 4>
264
Chapter 4
Figure 4.5.2
+"
= 7>
CONNECTIVITY and TRAVERSABILITY
= 4>
FACTS
'
-;70 " ½ E ( ! # E
= > = 4> "
'
-( 3 0 "
/ / (=> => = > ?
' -8 8L: 3/0 " := > E = >
'
-A90 :
=
>
= > = > " = > = > " = > = > E ' -+930 " + - =+> ' -(370 : " # & + ! & ! - =+ > "
'
-8L/0 -( 2H :/0 "
- =+ > + ! REMARK
'
6 #
1
Powers and Line Graphs
! & = >
DEFINITIONS
' 6 ;= > ! ! ! ! ;= >
5 5 = >
Section 4.5
265
Hamiltonian Graphs
' !
#
!
' + 5 = ½ > 5 ' 6 ! = > ? = > ! = > = >
'
= # 5 > = >
"
FACTS
' -* F+740 : ! 6 ;= > ½
' -2* //0 : ! 6 ;= >
! = >
'
' '
-+ 90 " -(9 0 "
" -8930>
! Æ = >
; =
> ? ;=;= >>
= > =
Planar Graphs FACTS
'
-630 A = -6470>
' -2730 : ! ! " 5 ! 5¼ # ! = >=5 5¼ > ? 1
4.5.3 Extending the Classics Adding Toughness DEFINITION
' " # + + ! & &
, & =+ > + ! =+ > & 6 #
FACTS
'
-H930 :
:= >
6
266
Chapter 4
'
-8 $L/10 :
CONNECTIVITY and TRAVERSABILITY
'
-8L/10 : 6
:= > 6
! Æ = >
REMARK
'
G 5 & & ( & ? *! -8 8: L110 # =/ 6> 6 $ 1 !
More Than Hamiltonian DEFINITIONS
' '
2 2
' ! / # / E =! > ( # #
2
' = > B = > B
FACTS
'
=> Æ = > => := > !
-8( 2:/90 "
EXAMPLE
' 6 ! 6 FACTS
'
-8990 "
'
! = > ¾
-*/10 " := > # ! : = > = 4> # ( Æ = > = E > #
'
-*/0 " ? = 9 >
5 ( 3 9 ! =(> E =3> E
Section 4.5
267
Hamiltonian Graphs
' -*/0 : / " ? = 9 > Æ = > / = > $ ¾ / E /¾
' -J /70 -J /30 6 # Æ = > = E > ! ' -J //0 : " => E # ' -( 2J: 0 : ! " => E => E = /> 5 REMARK
' 8 !
N# . ! ;
4.5.4 More Than One Hamiltonian Cycle? A Second Hamiltonian Cycle FACTS
' A 6 = -6 70>
' -6/30 "
/ ! / 11
! ! : # ! ! '* =! > = - => = =! >>> 6 ! ¼ ! ¼ ? ! # ! ! ¼ ! !
' -6/90 :
' -*110 ( # =>
! Æ = > => Æ = > Æ E " =Æ = >Æ = >>
' -$ 970 -2$ 970 6 #
! 5
!
Æ =
>
' -Q 970 -3/0 6 # , # 4
= > !
268
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARK
'
( 6 -6930 # % 5 ! 5 = ! > 6 # =( >
Many Hamiltonian Cycles FACTS
'
-6/70 :
! ' (½ 3½ (¾ 3¾ ( 3 (½
= > " 3½ 3 (½3½
= > " 3½ 3 $ ½ ¾ =1 ' > ! 3 ½ (3 ·½ = '>V ½
'
71¾ :¾ = > E
-( 340 :
= > "
5 = > " 7 ¾ ½ E 5
' -A /0 : ! = > " ! :¾= > Æ = > 5
Uniquely Hamiltonian Graphs DEFINITION
'
#
FACTS
' -A!310 6 # , B !
'
-H +3/0 B # = E /> B #
'
-8H /30 A B #
¾ =3 > E ! ? = ¾ > ½ ( B !
Section 4.5
269
Hamiltonian Graphs
Products and Hamiltonian Decompositions DEFINITIONS
' = E >
'
A ! & # = ½> = ¾> 6 ? ½ ¾
=
> ? =½ ¾>=½ ¾> ½ ? ½ ¾¾
=
6
=
> ? =½ ¾>=½ ¾> ½½
?
=
¾> ½ ½>
¾ ? ¾
¾
½½ =
?
¾ ¾
=
¾>
¾ = > ? =½ ¾>=½ ¾> ½ ? ½ ¾¾ ¾ ¾ ? ¾ ½½ = ½ > ½ ½ = ½ > ¾¾ =
6
½>
½
¾>
6 = ! > ? ½- ¾0
=
> ? =½ ¾>=½ ¾ > ½ ½
=
½ ? ½ ¾¾ =
½>
¾>
REMARK
'
H & -H 9/0 5 E B ' " ½ ¾ ½ ¾ T
FACTS
' -/0 : ½ ¾ ! 7 & ! & 7 6 ½ ¾ ! '
7 & => & =>
=>
¾
= >
½
'
77&
" ½ ¾ * ½
'
¾
-8/10 -Q3/0 ½ ¾ " ½ ¾
270
Chapter 4
CONNECTIVITY and TRAVERSABILITY
'
-( : /30 6 ½
'
-8 30 6 # !
' -J/90 = > "
=> =>
= > - 0 " = > - 0 " = > - 0
=> " = >
= > - 0
= > - 0 " = > - 0 " = E > = > E - 0 " # = > - 0
= > " =>
#
4.5.5 Random Graphs + 5= > ! - ? DEFINITIONS
'
=3 > 1 * : - ! * ' =3 . ( > 4 ? 4 = > ! &
6 7 ? @ ! 4 # + ! 7
' ! @ B = >
& & ? 1 -
' " Z ! Z " 5="> F B
"
' 6 4 2 %
# 2 6 %
Section 4.5
271
Hamiltonian Graphs
FACTS
'
- 970 -J970 6 # ¾ ' -J970 -J30 * ? E E 4 = > ? E E 6
'
4 = > ? = E
1 E > ' 5= > ? ' ' ' -+/ +/ 0 ( 5 5 ' -(/ 0 % ' -(110 % " % ; % % -J30 (
REMARKS
'
" - V =! B > ! C D = *> AI ! , 5 ! & 6 ! , 8 G
' " & ! # !
, 8 G
( ( -8((340
'
B
4.5.6 Forbidden Subgraphs DEFINITION
'
= > 6 - # 5 ! = > 6 => 6
; ! # 5 5 # # = &>
Figure 4.5.3
!" - ;
272
Chapter 4
CONNECTIVITY and TRAVERSABILITY
FACTS
'
-.2H 30 "
½ -
= >
= >
'
U
-8.J110 6 # ,
-
'
-8L/10 "
'
-2H 30 "
'
-8/0 "
'
-( 2 /40 "
-
-
-
1
Other Forbidden Pairs B ' T 6 ! -8/0 -( 2/90 1 + !
FACTS
'
,
?
+
+
6
, +
-
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275
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277
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CONNECTIVITY and TRAVERSABILITY
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293
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Traveling Salesman Problems
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299
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Chapter 4
CONNECTIVITY and TRAVERSABILITY
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305
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307
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COLORINGS and RELATED TOPICS
' , ? - : ; - ! , (' - + , . )
, 0
< :
1
:
;
! -, 0 ½
, ?
! 0
:
+ + " ,
." 0 J." 0 ." 0
¼¼
.0
¼¼
¼
: %& ./A A ; A " N F0 # - + , - " ¼ .0 < ¼.0 : .- EL0 E ? &+ ' - ,!'
, - " .0 < .0
O : ." + 8 2 9
A 4-+ E @22 -2A F 4 0 # -
+ " .0 < ¼¼
¼¼
.0
: .8 2 9
0 E ? +
' ' -
"
.0
0 !+ !! ' + !! REMARK
:
; , - /K + ' ' J.0 J.0C+ - F + 2 - ' ' 1 L 2 , - ) @- ' , ' ' ' + + 38 76 3 E*@ 5B6
FACTS
: 3 B 6 (' - ? . &$0 '+ 2 ' B
:
3 B 6 (' !V , ? . 0 '+ !! 2 ' &
:
38 5$6 # - (
( !! 2 '
Uniquely Edge-Colorable Graphs DEFINITION
: ; < . 0 ' ? ! , < ¼ .0 . ' ' 0 EXAMPLES
:
1 ! 1 ! - 1 !! . / - ' < 0
:
31&$6 1 ' , 5! 2 2 2 3 3 3 ' 23 . 5+ 50 !! 1 2 , !' ! ' <
FACTS
:
31&B6 1 ! + '
: (' ! ! + ' , =YIJ > ! ! D
Section 5.1
355
Graph Coloring
* - ½ ¾ ½¿ ¾¿ ' <
' - ,
, -
½ ¾ ¿
Further -Bound Graph Classes CONJECTURES
:
:
/+
.N 'N A @ 0 # -
'
3B&6 1
/
! '
! ' ' ' , -
FACTS
:
38S5 6 ;
! ' '
'
:
/
3@ 5&6 # -
/ + '
/ + ! ' ' /
- '
5.1.6 Coloring and Orientation Paths and Cycles FACTS
:
3$B+ * $&6 ;
:
3E$6 ;
:
'
'
.0 ' ' ' ' '
-
,
0 ' -
.
,
'
3156 ( Æ E K - ? ' ' .
0
.1 I* 1 +
0
E - -+ '
! ' ' ( ' +
-+ + '
:
!
3" &$6 ( - '
'
! . 0+
Eulerian Subgraphs DEFINITION
4 < . 4 0 4 < . 4 0 ' 4 , - + , .0 < .0 ' " 4 A + 4 < , - 4 .@ - ' " :
;
0
'
·
356
Chapter 5
COLORINGS and RELATED TOPICS
FACT
4 3;156 (' ' ' , - ' L ' ' , ' 4 ! ,- ) .0 ' + .1 - 0
:
Choosability and Orientations with Kernels FACTS
4 < . 4 0 - "4 ! @
* ' - * ' "4
* (' , .0 ' +
:
:
1 - - ; , * . % 0 ' ! % 3 " . 0 ( + Æ , !
: (' - ' - + , ! + ' + - 8M IF ( + ' ) -! + " .0 C E - ! . 0 Acyclic Orientations DEFINITION
: . ' 3E1 B 6 3;BB+ 60 1 , ( + S ; .Q;K0 ! ! ' + S @ .Q@ K0 +
1 - , ? ' 1 ' 5
) ', + , 6 2
9
#.0 ' ,
5 6
FACTS
:
3;115 6 (' < . 0 $ - , #.0 < +
& ' ' .(
2 ,+ ,-+ , #.0 C . 0 '
- 0 : 3;115 6 # - (+ < . 0 , ( #.0
+ '
Section 5.1
357
Graph Coloring
: 3;15 6 # , - + #. 0 < R. 0 , .0
,- 7 ) # + #. 0 < . 0
5.1.7 Colorings of Infinite Graphs FACTS
' ' - : 34 5 6 # + ) " .0 ' ' - ) ' !
: 3" &&6 (' .0 < + ' - ) 5 + :
3" 6 # + ) ) ' !
,
:
.0
5
38 56 1 .0 ,!) ' - + - !
9!% 1 A - + - ; '
:
38 BB6 1 ' , : 1 < + .0 < + ." 0 ,- ." 0
< . 0
Coloring Euclidean Spaces DEFINITIONS
:
1 ' - A
' ,
: - .) ) 0 = > 7 < + .70 - A , - ? ' ' 7 FACTS
: : : : : : :
&
3F 6 ( + . 0
3E E $6 " ! - - + . 0
3 76 ( ! + . 0
; ! 5 - . 0 3D* &6 ;
+ . 0 . C .00
3#9B6 ; + . 0 . C .00 3@ 5 6 1 " . 0 ) ' ' <
< .() , ) - 341 5 60
: # - ) 7+ ..700 7 C : # - 7 , 7 < + ..700 .3T760 # +
,-+ ' !
/ '
358
Chapter 5
COLORINGS and RELATED TOPICS
: 3 @2B 6 (' 7 ' + ..700 < E -+ ..7 00 < ..7 00 <
: .W 8 / A 3*1 76A W S 9 @ 0 D 7
' - + ' ' 7 < ' (' 7+ ..700 ) '
7A ' < 7+ 7 ..700 <
References 3;BB6 E ;+ 4$ " + 9I1 + 5BB 3;115 6 E ;+ 1 + T 1/+ @ ' ' + * !"5 .55 0+ I7 3;?8 @/B76 E ;? + 4 8 N
+ @/ N+ ;
* + .5 4$ 5 " 5 5 .5B70+ I$7 3;56 ; + * ' + I 8 92 . 0+ 4$ + S " ' + D E @ D
@ B&+ - S+ 55 3;776 ; + + 5 5 $ .7770+ $ I$B 3;8@556 ; + E 8-- + " @2 -+ D ' ! + 4$ 5 .5550+ I & 3;156 ; E 1+ ' + 4$ .550+ I 3;15 6 ; T 1/+ 1 + 3 5 5 $ .55 0+ $I$B 3"N &B6 ( "NN+ ;
' ' 8K ? + .5 4$ 5 " 5 .5&B0+ I$ 3" &&6 " " N+ + (5 !"5 5 5 .5&&0+ 5&I5B 3" &B6 " " N + + .5 4$ 5 " 5 ( .5&B0+ I$ 3" BB6 " " N + 1 ' + 4$ B 5BB+ 5I 3" B76 " " N+ S ; + S H + F,K ? ' - + ) 5 .5 4$ 5 .5B70+ 5 I55 3" &$6 4 ; " + ? ' D + .5 !"5 5 .0 .5&$0+ &&IB
Section 5.1
359
Graph Coloring
3" 8 &&6 % " ; 8 2+ % ' K + K + .5 4$ 5 " 5 ( .5&&0+ &I 7 3"76 @ " + ; ! ' !' + * !"5 .770+ I $ 3" 6 * D "
2+ % ' , 2+ # 5 4$ #"5 5 & 5 + 5 I5& 3" 6 "? S H + ; ' ) ' + & 5 %5 , "5 # 5 5 .5 0+ &I& 3 &B6 S ; + ; ' + * !"5 .5&B0+ BIB 3 &56 S ; + F?N
K ! ? : - + .5 4$ 5 " 5 ( $ .5&50+ $BI& 3 E*@ 5B6 ; - + E E + " *+ # @ + ; - 2 , : + + .5 " " B .55B0+ &IB$ 3 B 6 ; + %
- !! + S 1+ - ' 9
+ + 5B 3 $56 S + % + .5 43 $ 5 " $ .5$50+ &I&B 3 76 + ; ! ' !
+ * !"5 $ .770+ BI57 3 B6 " + @
2 5+ ) % 7 .5 B0
3 6 ; + ; ' ! 2 + .5 !"5 5 & .5 0+ B I5 3 &6 ; + ; ' * D "
2 ? ' F F,+ # 5 !"5 5 162 & .5 &0+ $I5 3 @2B 6 * "
+ S H
+ 8 @2 + + .5 5 .5B 0+ B$I77
4$ 5 " 5 (
3 5$6 E D + D ' !' / + 4$ $ .55$0+ I 3 56 S H + + 4 5 .5 !"5 .5 50+ IB 3 $&6 S H + @ 2 + 4+5 !"5 $ .5$&0+ I $ 3 F$$6 S
H
; F?+ % ' ! +
& .5$$0+ $I55
360
Chapter 5
COLORINGS and RELATED TOPICS
3 *1&56 S H + ; D * + F 1 +
+ S 9 ! ' + 1 + ; + ! ' + 4 5 & 5 XX( .5&50+ I & 3#9B6 S #2 * E 9 + ( , ! + 4$ .5B0+ &I$B 3$6 1 + 8 (+ #$5 !"5 7 5 5 5 5 B .5$0+ $ I5 3$B6 1 + % + IB S H
% F 8 . 0+ " " + E @ 4 " + 1 .F0+ 5$$+ ; S+ @ + 5$B 35 6 # -+ 1 ' + .5 4$ 5 " 5 ( $ .55 0+ I B 38 56 # - S 8 ?N + ' + # !"5 5 .550+ &I& 3 B 6 D + ; ' - ? + I$ + !F E @ + 5B 35$6 @ -+ ; F?N
!2 ' + * !"5 .55$0+ 55I7 3D & 6 , D D -N/+ ; ' + !"5 5 5 5 .5& 0+ I 7 3B&6 ; N'N+ S ' , ' + X(X .5B&0+ I M 3F 6 F F,+ 8)2 @ 22 + 8" "5 & "5 5 9 " BB .5 0+ I M 3F 6 F F,+ 2 2 * 2 E+ # 5 !"5 .5 0+ BI 3F@/&76 ; F? @/ N+ S
' ' ? ' H
+ $7I$ S H + ; *N+ 1 @N . 0+ 4$ " 5 77 + E @ 4 " + !F + 5&7 M 3F$6 F?N
+ 8 2 !'M + ,5 5 ! 3 " 3 5 3, $ !"53& 5 " + 7 .5$0+ $I& 341 5 6 1 * 4 " 1 ' + " 4 # $+ 9!( + 55 34 5$6 ; 4 + ; - ' ' + E + 4 55$ 34 5$6 ; 4 + 1 ' + S - + ; 55$
Section 5.1
Graph Coloring
361
34 5 6 S 4 + 1 ' + $ .55 0+ I B 3856 4 8+ !? , C . 0 + .5 4$ 5 " 5 5 .550+ I5 385$6 4 8+ ;
! + .5 4$ 5 " 5 & .55$0+ I 5 388 6 4 " 8 D E 8+ S + &$ .5 0+ &B$I&5
38S5 6 F ; 8 @ S + * , ' ! + .5 " " B .55 0+ 5I5 38@ B6 F ; 8 4 F @ + @ ' /K
- ' + * !"5 .5B0+ &&IB 385 6 4 F 8 + % "
2K ' + 4$ 5 # $$5 45 .55 0+ 5&I 38 5$6 E 8 + @2 , + .5 4$ 5 " 5 ( $& .55$0+ I & 38 76 E 8 + @ ' , ,!/
!V ,+ ) 5 .5 4$ 5 .770+ BI7$ 38 BB6 S 8 ?N + ) + 7 .5 !"5 $ .5BB0+ B I5 M 38M
$6 8M + ;, ' E+ !"5 5 && .5$0+ I $ 38 &B6 ; 8 2+ + + $&5I$5$ ; F?! 1 @N . 0+ 4$ + E @ 4 " B+ 8/ .F0+ 5&$+ !F + 5&B 38 B6 ; 8 2+ 1 F, ' , - ' - + ! *% 5 5 B .5B0+ &I B . *0 38 B 6 ; 8 2+ % F, ' + &I ; F?+ D D -N/+ 1 @N . 0+ - 7 : + E @ 4 " &+ .F0 5B+ !F + 5B 38 E&&6 ; 8 2 S E/ -+ ; ' + ! *% 5 5 7 .5&&0+ I5 . *0 381 556 4 8 -NY+ T 1/+ E + , ' :
+ BI5& * D . 0+ 4 * !" + (E; @ @ E 1 @ 5+ ; E @ + 555
362
Chapter 5
COLORINGS and RELATED TOPICS
38776 E 8-- + 1 ' + 4$ 5 # $$5 45 5 .7770+ 5I$ 3D* &6 D ; * + 1 / ' , + !"% 5 .5&0+ I 3D&B6 4 D, + - - ' , ' + * !"5 .5&B0+ $I$B 3D $B6 D D -N /+ % ' ) ! + !"5 5 5
5 5 .5$B0+ 5I$& 3D &6 D D -N /+ ( + 5 !"5 3 5 B .5&0+ $ I$B 3D &B6 D D -N/+ 8K ? + +
.5 4$ 5 " 5 .5&B0+ 5I 3DS@BB6 ; D
/2+ * S + S @2+ * ? + 4$ B .5BB0+ $I&& 3E1 B 6 E E 1 + 1 L ' ' F
' - + # 5 ; .0 < 2 (.0 < 3 ' (' (.0 < ' - - + , EXAMPLE
: # - $ + ' .$ C $0!
.3;1 5&60 ' C FACTS
: # - + , -
.0 ( + '
' +
: # - < . 0+ , -
. 0 . 0
.0 < .0
.0 .0
: # - - + -
.0 <
'
.3 4 &B60+ ' .' $50 ) ' , ' .3#5 60
: 3*@5B6 # - $+ '
.0 $+ ' . ' F,K ? 0
: 3D & 6 # - + .0 . C .00 ; + - 0
." 0
." 0
- C .,
'
: 3*576 # ' ' + 1
? . '
0 + +
.0 J.0 C ' -
: 38776 1
' + + ' - 7 < .0 .0 " .0 . C 0 .0+ ' - : 3;1 5&6 # - + - ' 23 .2 30!
.0
: ."
2K 1 ' 0 3 *1&56 @ +
' 2 (' .0 ' ' - - + !
: $ % 31 5$6 # 5+ .& &0!
' & : 1 D * . 31 5&6
+
0 ' .& &0!
: 31 5$6 - !
.& &0!
+ ' - &
Section 5.2
367
Further Topics in Graph Coloring
OPEN PROBLEMS
: : " .
3 *1&56 ( - .2 30!
.2& 3&0!
' 3 *1&56 - '
" 0 " .0" ." 0P
& P
" ' - +
REMARK
:
; Æ - , S , Æ - , S ! ' ' - 31 5$6
5.2.2 Graphs on Surfaces F + ' ' + ' , ' ' + , - - ! ' + ,
' ' ' ' ;?
' ' ' A ' -
, ' ,2 DEFINITIONS
: :
; , -
; . ' 0 ' ' '
: ;
' - ,2 ' ' FACTS
: : :
.#- 1 0 3F:B576 - ! .# 1 0 3; F&&+ ; F8 &&6 - !
31:BB76 ;
! ' ' !!
:
3" &56 - ! '
, '
: :
.M
/ K 1 0 3 56 -
¿ !' !
3F:B5B6 ; ! ' ' ' - - -
: - !
.315 60+ ! !
! .3 560
: :
3815 6 -
¿ !' !
315 6 ; ' )- !
368
Chapter 5
: : :
COLORINGS and RELATED TOPICS
3;156 ; !
3 5 6 1 !!
¿ !'
3 F&6 - - , ,
: 3F56 1 ' , REMARK
:
;
' # 1 - 3* @@15&6A '
' , - ' 2 ,
Heawood Number and the Empire Problem DEFINITION
:
1 ! " ' '
". 0 <
&C
E + ' -
" . &0
-9B: -:9. 88GD % )
6A) "8 3 ) ) 2 3 + , ) )
Section 6.1
495
Automorphisms
+ , * & 2
( ( 3+ ,
1 ) ) % *
$ ( 68 4 *
2
2 *
61! 8 3 ( 2 * 68 )
9 + , ( + , 4 ¾
$ ¾ + , % (
%
6.1.7 Primitivity DEFINITIONS
2 3 ) ) 4 +
, ( +4 , 4 4 +4 , 4 4 I ( )
$
) -<
) - ) -(
< )
) -(
2 4
$
4 ) -( (
2 )
& EXAMPLES
2
+ ,
$ ( ) - ) ) &
2
( )
< )
+ ,
2 2
>& (
) ) - C ( &
& FACTS
2
& & ) - + , =
2
$
( : E + ,(
&
496
Chapter 6
ALGEBRAIC GRAPH THEORY
2
6 1""8 3 ) %
( ( ( 1
%
2
6K1"?8 3 ) & % ( % (
4 4
2
6K1!!)8 3 ) & + , 4 Æ )
& ) )
( ( )
2 61 B8 3 ) % ( %
+ , 4 ( ( & ) & ) ) # ( ) ) / /
2 6 $ 511 "?8 3 ) % ( ( &
6.1.8 More Automorphisms of Infinite Graphs . / % ) ) 0 ./ ( )
% ( % ) = 6= 8 $ ) ( ) %
( + , DEFINITIONS
2 3 & & + , 1 & & & & & & + , ) +& &, +& &,
%
$ : + ,
2
:
+ , ( % 5+ , 4
+ ,
2 $ ) & + ,( & ) ) & $ & & + ,( & &
2
2 + , + ,
+ ,(
+ , +, +, + ,
$ ) 9 ( > % ) 9
Section 6.1
497
Automorphisms
2
%& % ) + ,( ( + , 4 %
2 ) + , + , % ) 1 + ,
2 % ) % FACTS
&' ()"* + *,
2 6GL 82 3 ) + ) % , : % ( ( 9 ) 5 ( & 9 &
2
$ ( % $ 5+ , 4 B
2
6= 8 3
& &
+ ,
% % )
: 2
& & J % ) & & % 9 +% , &&
2
1 % ( 5+ , : )
% % ) + , I + , + ,
2
6= !8 > %& % )
2
6= !8 $ % %& % ) (
2
% ( ) % + , ) % + ) ) ,
2
6= !8 > % %& )
2
6= !8 $ % ( + , ( 5+ , 4 ( A ) 7 = K 6K"8( )
% %
2 2
$
( % ( 5+ , 4 ¼
5 $ ( 5+ , 4 + . / ) , $
( 5+ , 4 $ & ( 5+ , 4 ¼
498
Chapter 6
ALGEBRAIC GRAPH THEORY
2 65 ?!8 $ : ( + , ) + , % ) 2 65 ?!8 ) :
2 6A)1"B8 5
% $ + , ) " + ,( 5+ , 4 $ " + , ( + , + H , 6+ , 4
2 65?8 3
) ( % ( & + ,
) % ) & + ,
2 6= "8 >
9 &
( ( ) %
9 ( %
9 $ (
9 )
& EXAMPLES
2 ) ¼ ( C (
) E ( % 5 % (
%
2 3 ) ) ( ) $ % (
& = ( % (
& $ (
+ ) , 2 3 + , 4 ( + , +, , +, H , +, B, +, H B, , ( ) 7
: 5+ , 4
2 % + % ,
¼
2
> + , & : &( +
, : &( +
, & & : &
2 ) + (
&, & ( ) & Strips
Section 6.1
499
Automorphisms
DEFINITIONS
2
%+ ,,
+
2
& )
+ , %+ ,
+ ,
%
( ½ + ,(
+ ,
%
REMARK
2
$
(
) % %
½
(
$
9 +5 6K1!!8,
FACTS
2
7 % ( : 2
1- ( 1 1 ( E ) % ( # ! +?"?,( M " 6 1""8 K > # > 1- ( C %
( +?"",( ?M B 6 " 8 $ -( N % ( $% &' +?" ,( ?MBB L 6= 8 = ( ') 1 (
+?
,( M!
6= "8 = ( N #& C C 1 ( ( +? ",( ?M! 6= !8 = ( % % ( $ " +?!,( M"
502
Chapter 6
ALGEBRAIC GRAPH THEORY
6= "8 7 = > # ( E ( # 5 = +? ",( M ? L 6=! 8 N =C ( ') L N TL ) ( ! $ !+ ( ?!
) *
6= "8 N 7 = ( ) ( +?",( BMB L 6$ ?8 1 $ ( ') & - - ( ! +? ?,( "M
#
6$ !B8 1 $ ( ) ( 2 &! ! +I : # 5 KS A I : A U ( = ? ?,( > U ( ( P 5S ( J = ( ( ?!B( M 6$ !8 1 $ ( C - ( ( / &&( +?!,( BM?
+ ,! - .!!
6$ $C! 8 1 $ = $C) - ( ( +?! ,( !!M" 6$ G BB8 1 $ 5 G VC ( 1 D 5 ( $( J Q -( BBB
!
# !* % " ( K
6$ 5""8 1 $ J 5 ( ( +?""R"?,( M! 6$"!8 P $ ( E ) & ( +?"!,( !M" 6K"8 = K ( % ( " M"" 6K?8 = K ( E % %& % ( M
)!
! )!
+?",(
)! +??,(
6KJ ?8 = K J ( N % ( ( +?? ,( " MB 6K1!!8 = K # > 1- ( E % % ( ! +?!!,( M 6K1!!)8 = K # > 1- ( E % & ( )! ( M 6K1"8 = K # > 1- ( 7 % ( ! +?",( ?M 6K1"?8 = K # > 1- ( %
( +?"?,( M !
Section 6.1
503
Automorphisms
6GL 8 N GL ( ( 3 C ( ?
# ( -
P
6J "8 3 J C( E & C ( +? ",( ?M 6J 1!8 3 J C # > 1- ( ) ( $( +?!,( ??MBB" 6J 1!)8 3 J C # > 1- ( ) ( $$( +?!,( BB?MB" 6 1? 8 J # > 1- ( E ) ( " +?? ,( M
65 !8 5) ( ( +? !,( M 65 B8 5) ( (
0! +?
65 8 5) ( & ( !M !" 65 8 5) ( P & ( 65 !8 5) ( W (
B,( M !
) - +?
! +?
% ( J
,(
,( M"
( # ? !
65?8 J 5 (
( +??,( M
65 ?!8 J 5 P $ % ( : ( +??!,( B MB 61"?8 I # > 1- ( $% & ( ( ( +?"?,( "M 6 " 8 P $ % (
( $% - +?" ,( J ( BM! 6 8 1 (
# !( ' ( ( ?
61!8 # > 1- ( E ) (
+?!,( ? MB
61!8 # > 1- ( E 1( 2 # &! !( +Q ( N 3 -( 1 ( , 5 P ( A ( ?!( B M 61!8 # > 1- ( (
(
( 1 +?!,( BM B 61! 8 # > 1- ( (
+?! ,( !M
504
61?8 # > 1- ( > ( BM!
Chapter 6
ALGEBRAIC GRAPH THEORY
+??,(
61 B8 # > 1- K > ( C %
( ( L 61!8 # 1 ( ') L ( ?M
+?!,(
618 = 1 ( I ( +?,( BM "
Section 6.2
6.2
505
Cayley Graphs
CAYLEY GRAPHS I $
5) 7 C 7
Introduction :
: E
I )9
6.2.1 Construction and Recognition 1 % ( % ( ) ) % 1 I % I % ( I % I % ( DEFINITIONS
2 3 ) % 3 ) ) 4 ½ ( ( ½
( I+ < ,( %2 I+ < , < 9 2 < I+ < , < 4 2
1 : $ ) ) ( ) ( 4 < 4 2 H
I
I+ ,
2 I 1 I +< ,
2 2 2 2
) 2 +
: 7 +>,(
2
#
+ ,(
4
!
Figure 6.2.1
)
-
- "- .' $ *"" "
FACTS
2
> I &
2
I I+
2
65 "8
) / + >
/>
2
/ > < ¾ > 4 / ( > 4 < > 4 H / / 4 > 4 / 4 H < / 4 !( > 4
/
# + ,
/
4 !(
/
# > # + ,(
/
# > # + ,(
/
4 !(
>
/>
/ + >
ALGEBRAIC GRAPH THEORY
>
# + ,
>
4 <
) / + > + />
/>( / > ) ( />( / > ) ) 2 ·½ /> 4 + H ,+ H ,( $<
/>
/>
4 + H ,+ ,(
4
H ,)
/>
4 + ,)
4 +> H ,)
,)(
/
< / > /
>
4 +/ H ,)
>
4 + ,)
/
4 / H (
4 +' H ,)+' H ,(
'
/
/
4
'
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/
/ ( >
/ ( >
4
4 +' ,)+' ,(
) +'
4
H (
)
,)+' ,( 4 +' ,)+' ,(
)
4 ( 4 H ( 4
4
4 ?<
/
4 !(
>
4 !
4 !
RESEARCH PROBLEM
42 $ ) ' ( B ' Y - C )
6.2.3 Isomorphism 5 - I :
I
Section 6.2
509
Cayley Graphs
DEFINITIONS AND NOTATIONS
2
I I+ < , " I+ < , 4 I+ < ,( & %+ , 4 +,
2 2
" I
I$
3
2
3 4 /½½ /¾ / ) C 7 , B , + ( + , 4 +, , , ,
,
: $
,
# ,+ / , B ,
+ /
( 4 $
3 & +, 4 ,+, 2 B , + 7 ) & $( & +, & +, , 4 B " 1 ( & & 2 3 4 /½ /¾ / ) C $ 4 + , $
+ , J
4 B( 4 $
3 + , )
( & $ $ ( &( 4 ( ( +&, 4 +&,) $ ( +&, 4 +&,( +&,
&
EXAMPLES
2 I +!< , I +!< , - 2 2 N% ) ( )
2
7
4 ( 6 4
4
? ? B 8
? B ?
I + < , I + < , ) E (
4 % ( I$ FACTS
2 6A!!8 3 ) I % I$ ) + , 9 + ,
510
Chapter 6
2 63 B8 $
I$ (
ALGEBRAIC GRAPH THEORY
)
2 6#?!8 I$ 4 ,( :
B (
,
" ? "
2 3 / ) / I$
E (
/ 4
/ H
( I$ :
E I$ I I$ ) S * 6A 8 &
2 6 !8 $ / ( ) & /
;+, /
+/ ,) ; >
2 6 # B8 $
4 / / / ( / 4 ( )
/ / /
;+,
½ ¾
;+,;+ , ;+,
& Ê
- ) ( - ) & $ / 4 ( & 4
& - ) EXAMPLE
2 1 7 4 B ) +(,( +(,( +(,( +(,( +(, +(, ( & ( +(, 5 / 4 ( & 4 ( ) & 4 +, 4 7 & 4 ( +&, 4 & 4 - ) 7 & 4 ( +&, 4 & 4 "( ) ) 7 ( & 4 ( +&, 4 & 4 " ) 1 ) % B RESEARCH PROBLEM
42 7 /(
I$
Section 6.2
511
Cayley Graphs
6.2.4 Subgraphs : ) I 5
& (
$ 9 &
& )
DEFINITIONS
2
½
( (
2
)
(
( 4 ( ! "( ( ! " ! "
%&
)
! "
4
(
=
(
$
- +
2
0 4
)
4(
0
=
FACTS
2
3
$
2
)
(
$
& $
(
&
& (
2
6#!( 1!B8 $
&
2
6 ?8 $
I+
2
2
I
( =
/ ( /
512
Chapter 6
ALGEBRAIC GRAPH THEORY
6.2.5 Factorization DEFINITIONS
2 2
( ) ½ )
2 =
( =
2
) ) FACTS
2
65" 8 > I 2 4 '<
) <
C
2
65" 8 I
I C 2
I 2 $( )! +BB,( !M ? 6A!!8 3 A) ( $ )
( " ? +?!!,( ?M
6A? 8 3 A) ( ( ( (
- 2
! 3 &&( 3 ( # L 3 3 SC( #$ J = ( ?? ( !M B 6A 8 J A 9( S * ( I ( 1 ( J Q -( ?
!(
514
Chapter 6
ALGEBRAIC GRAPH THEORY
6IZ"8 I I I J Z ( E
) ( I ) # P$$$( ( ! ! ( 5 P ( ?"( M 67 ?B8 K 7 -( E ) ( +??B,( B M 6 "!8 K 3 1 - ( ?"!
#
" # ( 1 ( J Q -(
6$ B8 # $ I > ( E I & ( +BB,( M? 63 ??8 I = 3 ( 7 I$ ) ( ?M
63 B8 I = 3 ( E % I \ ( +BB,( BM
+???,(
)!
63 ? 8 K 3 ( =
I ) ( ( +?? ,( ! M" 63 B8 K 3 ( =
I ) ( ( 63? 8 3) C-( I 2 ( & -( ( +?? ,( M"? 6#!8 1 # ( > > ( +?!,( M S * 9 ( 6#?!8 # #C-( E S +??!,( ?!M B
!
)!
V 6 > B8 A ( K 5 S V( K9( - # 1- ( I ( 6 !8 (
( 5 P ( J Q -( ?!
65 "8 5) ( E %& ( +? ",( "BBM"B 65" 8 5 ( E C) I ( +?" ,( ?"MB!
6 !8 K (
) ( +? !,( M 61!B8 # 1- ( I ( M? 61B8 1 ( # ! 2 # ! = ( ( BB
2!(
+?!B,(
# 5 ( J
Section 6.2
515
Cayley Graphs
61 "8 N 1 ( I
(
6[B8 K [(
+?",( B!M
" !! 2 & (4 -!
( G (
N ( BB
6OB8 [ O( I ) 2 ( B
)!
+BB,( !M
516
6.3
Chapter 6
ALGEBRAIC GRAPH THEORY
ENUMERATION
! I I I I I
5 # N I
I 5
Introduction $ - I ) C :
(
- ) C
$( ( : & ) # : ) ?! - S + 6S "!8 > , 7 = 6= 8 & (
( (
) I 6I !( I"?8( % . / #
) A =C 6A =( A =)8
) S 6S "!8( E 6E"8( = 6= ?8 & ( ) )
( ) 6=!8 ( - % :( : - % 65 ? 8
6.3.1 Counting Simple Graphs and Multigraphs DEFINITIONS
2 ) ( ½ ¾ ( ) )
)
2 4
Section 6.3
517
Enumeration
2
. ) .
% ) 2 ' . +, . +, .
2
. .
)
FACTS
2
) ) , )
5 ) Æ ,
2
+ , 7 , ( ¾ ( ) ) , ) ) ,
2
) )
Table 6.3.1 ,
B
5 )
5$*" * - 2 " "
B B B B B B
(B
B
! " ? B
+¾ ,
"
,
!
"
B ( (BB (BB ( ( (BB (BB ( B
B (B (?" B(? ( ("B B(?B ?(?B (! (! ?(?B B(?B ("B
" !" (! B(! ?"("B ! (!B ("(BB (B"(B (?B (?BB ((B (!("B B((! !(( B B( ( BB
(! "
(B?!(
2 6 8 ) )
4
+¾ ,
4
+¾ ,
"( (
(
)
5 )
2 S * ( ( +* , 4 * )
518
Chapter 6
Table 6.3.2
ALGEBRAIC GRAPH THEORY
1 " *$*" * - 2
"
!"
! (!B
("
(
" ( "( ?
) - C ' +* ,( ' 4 7 & ( +* , !( * 4 ( * 4 ( * 4 * 4 * 4 * 4 * 4 B
2
5 + ,
( ( + , 4
]
] ' * ]
+ ,+
¾ ¾·½ ,
¾
= - +* , ( + , + ,
( >& + , 2
4 4 4 H ]
H
]
]
]
4 H ? H " H
4 H B H B H 4 H
H B H B H
H B H B H "B H B
H H B H B
2
) )
+¾ ,
2 +, 4
) ,
2
6= ( S "!8 2 +, & ) ) ) & + , ) ) H 5 )
2
) )
& + , ) ) ) 5 )
Section 6.3
519
Enumeration
Table 6.3.3 /* -
, B
! " ? B
2 "
, "
? ?
!
"
B ?! " " ?!
B ?"B ( ( ! (
(B
(
+¾ ,
2 ( ) % 2 )
, +, 4
)
)
, 2 6= ( S "!8 , +, &
) ) ) & + , ) ) % H H H H 5 )
9
½ 9
Table 6.3.4 5
* * -
, B ! " ? B
2 " , "
! " B
! ! ! ? ? ? ?!
" " ! B
" ! ( " (!
520
Chapter 6
ALGEBRAIC GRAPH THEORY
EXAMPLES
2 7 0 0 ) % ) B 0 )
Figure 6.3.1
/* - 2 " "
2 7
7 & 0
Figure 6.3.2
0"" * *
* * - 2 " "
6.3.2 Counting Digraphs and Tournaments DEFINITIONS
2 ) ( ½ ( ) ) )
2 + , ( ( ( & ) )
2 + , ( ( &
2
. ) . 4 % ) . 2 + , ' +. +, . +,
2
)
. . FACTS
2 ) ) , )
Æ 5 )
2
7 , ( ( ) ) , ) ) + , ,
Section 6.3
521
Enumeration
2 " ,
Table 6.3.5 5$*" *
* " -
,
B
B ?B (B (" ( B "(! B !!( B (?!B !(? B "(! (B"( !
B
B ? !? ? !? ? B
! " ? B
(B?
2 ) ) )
5
+¾ ,
2 ) )
)
(
2 5 + ,
( ( + , 4
]
] ' * ]
+¾ ,
= - +* , ( + , + ,
(
>& + , 2
4 4 H
]
]
]
H " H H
]
]
4 H H 4 H
4 H B H B H H B H B H
4 H H B H
H ?B H B
H H H ?B H B H B
522
Chapter 6
ALGEBRAIC GRAPH THEORY
2 +, ) )
+, 4
) ,
2
6= ( S "!8 +, &
) ) ) & + , ) ) H 5 )
2
) )
& + , ) ) ) 5 )
Table 6.3.6
- 2 " ,
,
B
! " " " !
!? !B! ( (?B ( !B
"
?( B"
! " ? B
2 2
( ) %
´ ½µ
6N 8 ) " ) ] " 4 ] ' * ] +* , C ) -(
+* , 4
+ ,* *
*
5 ) !
2
6# "8 3 " +, 4 H H H H H H ) ( 7 +, 4 H H H H H )
" +, +, 4 H " +, 5 ) ! J &
Section 6.3
523
Enumeration
Table 6.3.7
" - 2
5
(""B ?( ?(!(B ?B(! (" (B"(( "
B
! " ? B
(BB" !"( ?( (?? ""( ( ?B (B(?(!
EXAMPLES
2 7 0 ) % ) B 0 )
Figure 6.3.3
*
* " - 2 "
2 7 E
Figure 6.3.4
- 2
6.3.3 Counting Generic Trees DEFINITIONS
2 ) ( ½ ¾ ( ) ) ) )
2 &( ( %
524
Chapter 6
ALGEBRAIC GRAPH THEORY
2 + , FACTS
2 1%*% * 6I"?82 )
5 ) "
2
) ) 5 ) "
Table 6.3.8
5$*" "
" *$*" - 2
3)
3)
? !(!! !( ? (B?!( (B (! (BBB(BBB(BBB (?!(( B !(BB"(!B( "" (?"(B" ((" !?(!(!!( ( ?(?(? (B (?B( ( (?( B( B (" (?"B
! " ? B
2
)
(? ("B! ( (!"(? ? BB(BBB(BBB ( !(?!( ? (?!( ( (!?( B(?(B! ( ?(?(! (? (? (? (B "( ?(! !(B !( ?(B!(?!(?
) &
+, 4
½ &
4 H H H H ? H B H
2 6I !8 Æ & +, )
+, 4
½
+ ,
½
% & S + 6S "!8,
+, 4 & 5 ) ?
+ ,
Section 6.3
525
Enumeration
2 ) "
$+, 4
½
" 4 H H H H H H
2 6 * 6E"82 Æ " $+, )
+, 7 ! )
$+, 4 +, +, +,
5 ) ?
2 I : &
+, 4
½
7 4 H H H H H H B H
2 Æ 7 & )
+, 4
½
+ , H
% &
+ , 4
H
&
½
+ ,
2 ) !
J ) ( > ) ( 6# "8 K 1 # (
! !( = ( 1 ( ?
6E"8 E ( ) (
"
2 ? +?",( "M ??
6S "!8 S I ( !( 5 P ( ?"! 6 "8 K (
" +? ,( B M
2 # !* # !*
& !!( 1 ( ? "
65 ? 8 J K 5 5 0(
( ??
2 &" 1 !(
Section 6.4
6.4
533
Graphs and Vector Spaces
GRAPHS AND VECTOR SPACES " !# ! $ A I N% I 5) ' I 5) ' ) I I 5) I I 5 N I RI A ! C I I 5
Introduction > )
) ) 2 G 0* ( G 0* E * > ) ( ( ) ) G 0* : ) C ( G 0* : ) C ( % ) ) ( $ ( ( 9
5
) % ) ) ) & ) T # )
C
6.4.1 Basic Concepts and Definitions
( ( )
) I 7 - ( ) ) % 7 ( 6 Q??8 65?8
' %(
(
4 ½ ¾ (
,
(
4 + , + , 4 ½ ¾
534
Chapter 6
ALGEBRAIC GRAPH THEORY
$ + , ( ) ( ) + , DEFINITION
2 &
) ) ^ REMARK
2
$ + ( ,
EXAMPLE
2
>& ? 7
v1 e1
e3
v2
v3 e2
e4
e5
e7 e6
v5
e8
v4
Figure 6.4.1 Subgraphs and Complements DEFINITIONS
2
4 + , 4 + ,
2
> ) % : ) 4 + , 4 + ,( ) % J )
2
> ) % : ) 4 + , 4 + ,( ) & J & )
2
) 4 + , 4 + ,( ) , + ,
4 +
Section 6.4
535
Graphs and Vector Spaces
EXAMPLES
2 7
4 ½ ( )
7 7 +, 7 4 ( & ) 7 +),
v1 e1
e3
v2
v3
v5
v4
e8
(a) An edge- induced subgraph of the graph G
v1 e1 v2
v4 (b) A vertex-induced subgraph of the graph G
Figure 6.4.2 0 "#" " $ " 23#" " $
2 )
7 +),
7 +,
v1
v1 e3
v2
e1 v3
v2
v3
e2 e4
e5
e7 e6
v5
v4
v5
(a) Subgraph G'
Figure 6.4.3 0 $
e8
v4
(b) Complement of G' in the graph G
" *
7
536
Chapter 6
ALGEBRAIC GRAPH THEORY
Components, Spanning Trees, and Cospanning Trees DEFINITIONS
2
-
( N% B
2 &
5 $
(
% N% !
2 2
)
& ) & )
2 ) $
"
2 / (
/
2 "
"
2 3 ) & , / 8+ , + , ) 8+ , 4 / + , 4 , H /
EXAMPLE
2
" 7 v1 v1
7
e1
e3
v2
v3
v2
v3
e2 e4
v5
v4 (a) A spanning tree T of G
Figure 6.4.4
e5
e7
v5
e6
e8
v4
(b) The cospanning tree with respect to T
0 " "
FACTS
2
& )
Section 6.4
537
Graphs and Vector Spaces
2 & ) , H / / ) , H / REMARK
2
' (
Cuts and Cutsets DEFINITIONS
2 I 4 + , 3 ½ ) 9 ) 4 + ( , & ! " & (
2 $ (
( >: (
EXAMPLE
2
7 7 ( ! "( 4 4 ( ( 7 +, 5 ( ! " (
7 +),
v1 v3
e1 e2
v2 e4 e7
v5
v4
e8 (a) A cut of the graph G
v1 v2 v3
e5 e4 e6
v5 v4
e7 (b) A cutset of the graph G
Figure 6.4.5
0 "
538
Chapter 6
ALGEBRAIC GRAPH THEORY
The Vector Space of a Graph under Ring Sum of Its Edge Subsets DEFINITIONS
2
5 4 ½ ) ) )
7 & ( ) + B B B B, ) 7
2
+ ' , ( ( ) ) )
2 , 4 + , 4 + , 4 +? ? ? ? ? , ( ? 4 ) ) ( + + ( ) B 4 < B ) 4 < B ) B 4 B< ) 4 B, FACT
2 , ) , ) , ) + , , 7+,( % (
) + ) , ) _+ , REMARKS
2 2
)
$ ) ) , + , ) ) , ) E) + ^, B _+ ,
2 2
$ ( 6 8
) & 65 8( 6I!)8( 6N!8( 65?8( 65"8
6.4.2 The Circuit Subspace in an Undirected Graph DEFINITIONS
2 & I (
2
) ` + , $ ( ` + ,
9 ^,
+
Section 6.4
539
Graphs and Vector Spaces
FACTS
2 2
)
9 ( ) ) )
2
) ( ` + ,
2 ` + , ) _+ ,
EXAMPLE
2 7 ( ( 7 v1 e1
v1 e3
v2
e1
e3
v3 v 2
v3
v2
e2
v3
e2 e4
e7
e7
e4
e6
e6 v5
v4 v 5
e8 (a) Circ G1
Figure 6.4.6
e8
v4
(b) Circ G2
v5 (c) G1
G2
- "
REMARKS
2 2
7 ) P) 6P8
( + Ü,
( ( &
Fundamental Circuits and the Dimension of the Circuit Subspace DEFINITION
2 " : (
$
" (
FACTS
2
" ( , H ( "
540
Chapter 6
ALGEBRAIC GRAPH THEORY
2 " ( (
"
2
+, H , ` + , )
2 $ ( ) &
2
) ) ` + ,( ( ` + ,
: , H ( + ,
2
) ` + , : + , 4 , H /
/
EXAMPLE
2
7
"
4
½
I I I I
4
4
4
4
$ ) % ( ( ( ( ( 7
6.4.3 The Cutset Subspace in an Undirected Graph % 5 ) ) & DEFINITION
2 9
) 6+ , ^ ) 6+ , FACTS
2
> 9 ( 6+ ,
2 2
6+ , ) _+ ,
< ( 6+ ,
Section 6.4
541
Graphs and Vector Spaces
EXAMPLE
2
I 7 ½ 4 !½ " 4 ! " 7 ( 4 ( 4 ( 4 4 4 ( 4 ( 4 # ( ) 4 !0 4 "( 0 4 4 ( 4
4 4 (
4 4 (
4 4
$ ( ) 7 Fundamental Cutsets and the Dimension of the Cutset Subspace DEFINITIONS
2
3 " ) ( ) ) " $ & " ( ! "
) "
$ ) )
2 &
"
(
FACTS
2
( ) "
"
(
2
) " ) ( ( ) ) "
2
& ) 6+ ,
2 $ ) ( ) & 5
2
) ) 6+ , ( 6+ ,
: ( - 8+ ,
2 ) 6+ , : 8+ , 4 /
/
2 & ) ) 6+ ,
542
Chapter 6
ALGEBRAIC GRAPH THEORY
EXAMPLE
2 7 7 ( " 4 ½ A A A
4 (
4 (
4
,(
A 4 $ ) % 4 ) ( ( ( ( ( ( 7 B
6.4.4 Relationship between Circuit and Cutset Subspaces A ) ( 7 7 5 - ) & & Orthogonality of Circuit and Cutset Subspaces DEFINITIONS
2 ) , < ) < ,
,
2
)
+ , C J C ) ) FACTS
2
)
=( )
2 7+, C
2 ) ) ) )
) ) >: (
2
) ) ) )
) ) >: (
Section 6.4
2
543
Graphs and Vector Spaces
)
Circ/Cut-Based Decomposition of Graphs and Subgraphs DEFINITION
2
)
) & J C
FACTS
2
$ ) (
2
6I!8 )
)
2
$ ) ( ) + , ) &
2
6I!)( 1 #!8 > ) $ ) : '( EXAMPLES
2
I 7 ! $ ) % ) ) 5 )
) _+ , E ) ½ ( ( ( 2
4 + B B B
B,
4 +B B B
B,
4 +B B B
B ,
4 +B B B
B B ,
4 + B
B B,
4 + B B
B B,
4 +B B
B B ,
$ ) ) & ( 7
544
Chapter 6
ALGEBRAIC GRAPH THEORY
7 ( +B B B ,( ) ) ( ) & 2 +B B 4
+
B ,
B B B
4
B B, + B ,
+ B B B B B ,
( + B ,
e4
e7 e3
e2
e5
e6
e1 Figure 6.4.7 9
**
2 I
7 " $ ( ( ( = ) ) ) & = ( 7 ( ) % 2
+ ,
4
+ B B
B, +B B
+ B B B , ( ( ( (
B
( +B B B ,
v1 e1 e6 v2 Figure 6.4.8 9
e4
e2 v4 e5 e3
,
v3
**
Section 6.4
545
Graphs and Vector Spaces
6.4.5 The Circuit and Cutset Spaces in a Directed Graph $ ( A(
&( 0
:
$ (
)
&
( (
)
( ( ( (
Circuit and Cut Vectors and Matrices DEFINITIONS
2
) ( -
- + ,
2
3 ) 4 + ,
(
% )
2
+ , ) (
2
( 4 + ,
% )
2
3 ) 4
½ ¾ (
)
,
+½ ¾ ,(
4
2
B
3 ) 4
½ ¾ (
)
,
+½ ¾ ,(
4
2
B
3 ) 4
½ ¾
3 ½ ¾
½ ¾ ) ( (
$
,
&
546
Chapter 6
ALGEBRAIC GRAPH THEORY
,
&
The Fundamental Circuit, Fundamental Cutset, and Incidence Matrices
J&( % & REMARK
2
% % ) ( ( ( & + ( ,( & ( & ) +½ ,(
4 B
+ ,
+ ,
DEFINITIONS
2
) " ( ) ( ( +, H , ) & & 5
( & " ( ) 1 ( + , ) & & 3
"
2 ( 0 ( ) & & ) & & ) 0
2 & )
: ) & & : ( B EXAMPLES
2 I 7 ?+, ) 7 ?+), +,( + B B B , + B B B B ,(
2 I " 7 ?+, ( ( ( " &
& 2 7 I # &2 I I I
B
B
B
B B B B B B B B
Section 6.4
547
Graphs and Vector Spaces
v5 e4
e7 v1 e3
v3
v2
e1
v1
v4 e2
e2
e2
v3
e6
e5
v4
v2 e5
e5
v5
e6 v3 v1
e3
v4
v2
e1 (a) A directed graph.
(b) A circuit with orientation
(c) A cut with orientation
Figure 6.4.9 0 " " 7 7 " - 7
I # &2 A A A A
$ # &2 J J J J J
½
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B B
B
Orthogonality and the Matrix Tree Theorem
)
)
FACTS
2
)
'
$
( )
'(
( C
( + ,
3
)
) +,
,
)
% ) ` + +),
% )
6+ ,
)
548
Chapter 6
2
$
2
ALGEBRAIC GRAPH THEORY
% $ ( ) &
) Æ
)
) &
5
(
)
Æ )
2
` + , 6+ , $ (
(
2
` + , 6+ ,
%
) ) ( : ,
H /(
( :
/(
- ( / )
2
)
2
2
I "
½
)
)
4
& 4
& C ,
5
4
H 4
C
1
1
4 6@
) & 1
2
1
@
4 64
( 8( @
) & 4
) & 1
8(
" 5
(
@
&
" # ( 1
4
4
&
) 5
( &
2
+
( ! !
& 0 0
, 7 (
: )
EXAMPLE
2
>&
7
REMARKS
2
A
)
% (
2 +,
- H ,(
+ N% ,
& +
% -
,
Section 6.4
Graphs and Vector Spaces
549
2
& 0 0 ( $ ) % + , & : & 0 + * , : ) & & + , 7 ) 65?8 &
65"8 Minty’s Painting Theorem
+ , ) DEFINITIONS
2
2
2
&( ( 4 1 C ( ) ( ( ) ( & ) - FACTS
2
+ ! , 6# 8 3 ) 7 ( & 2 & - ) ) ( - & - ) ( -
2
> ( ) ) )
REMARK
2
# * + - , )
+ 65?8, 7 7 E - ) # ) % 6# B( # 8 5
) ) 6PI"B( I ! ( 1 !B8
6.4.6 Two Circ/Cut-Based Tripartitions of a Graph $ Ü A ) 9
550
Chapter 6
ALGEBRAIC GRAPH THEORY
1 ) ) Bicycle-Based Tripartition DEFINITION
2 ) ) ( ) EXAMPLE
2
½ ( ( ( 7 " )
FACT
2
6 !"8
2
)
)
)
% ) 7
REMARK
2
6 !"8 7 ) 6?8
A Tripartition Based on Maximally Distant Spanning Trees DEFINITIONS
2 ( +" ",( ) % +" ", 4 +" , +" , 4 +" , +" ,
"
2 " " +" ", " "
+
"
" "
,
& ) )
2 & " " 5
" " ) 2
3 A ) " % ) 3 A ) " % ) A
Section 6.4
551
Graphs and Vector Spaces
) ( ) : A½ ( A A 4 A ) A + & -
2
) N%
)
2
+
& " " , ' )
) ' )
) FACTS
2
6G G ?8 3 " " & " " % )
" "
) " " % )
) " "
2
6G G ?8 I 4 + , 3 ) ( ( 4 + , + , ( ( + , + , &
EXAMPLE
2 $ ) % " 4 " 4
& 7 B 2 4 ( 4 ( 4
e1
v2
v3
e5 v1
e8 e4
e2
e6 e7
v4
e3
v5
Figure 6.4.10 REMARKS
2 $ ) + - , : , H ) ) ) )
, H
) : ) :
552
Chapter 6
ALGEBRAIC GRAPH THEORY
) ) ) C : ) ) , H ( - E- ( $ C- ( 1) 6E$1!B8 ) ( & ) ( (
) ) : ) ) )
- ) G G9 6G G ?8 ) E- ( $ C- ( 1) 6E$1!B8 ) - 5 65"8 & )
2 3 63 ! 8 A 1 ) 6A 1!8 &
6.4.7 Realization of Circuit and Cutset Spaces $ ) ( & -
&
& ) ) 6 ?8 I ) 6I "8 6 "8 ) ) 1 Æ C) & & 5 65 8
) ( & Æ * C) DEFINITIONS
2
7
& - +
v5
v6
v4
v1 v2
v3 (a)
Figure 6.4.11
2
v5
v1
v4 v2
v3 (b)
- & - +
& 7 7 4 67 @ 8( @ &( )
Section 6.4
553
Graphs and Vector Spaces
7
2 &
) Æ ( ( B 2
9
(
( B(
& ) + Æ ( ( B, C + , ) C
2
)
& ) * * -
FACTS
2
7 (
&
2 2
0
"
&
7
& ( ( B ( * ) * - ) + - & ,
7
2 & & (
B 4 B B
B
7
B
2
7
2
7
& C) & & G - 7
7
& C) &
& G - 7
7
REMARKS
2
# 6#!B8 * C) ( * ( !
2 > - % 6 B( 8 A 6A!B8 % & ) & & ) &
& C ) A * 65"8
2
& C )
- $ & I )
6I "( I ?8 C ) $ ( A 6A!B8 A Q 6A Q 8 C
554
Chapter 6
ALGEBRAIC GRAPH THEORY
& & N A * ) ) 65"8 Whitney and Kuratowski
1 ) - ) 1 618 G - 6GB8 1 ) ) ( 1 ) = % ) + 65?8, - ) N% : DEFINITION
2 ¾ ½ ) ¾ ¾ ½ ½ FACTS
2
$ ) ¾
½( ½
2 2
¾
618
$ -( G - 6GB8 ) G - REMARK
2
5 61B8 G - * $ : G - * C) & C )
References 6A!B8 P P A ( 5 2 (4 - &! !( N ( N > > ( $ $ ( # ( $ ( ?!B 6A Q 8 7 A N I Q ( 5 -( ! +? ,( M B
&
6A 1!8 K A 3 1 ) (
( " &! ! +!,( !M 6I "8 $ I ) ( # )
( B( ! + ",( M 6I ?8 $ I ) ( & ) - ( +? ?,( !M!
&% !
Section 6.4
555
Graphs and Vector Spaces
6I!8 1 G I( E ( B +?!,( M ? 6I!)8 1 G I(
# ( J =
&
( ( ?!
6I ! 8 3 E I N # (
-( & ! ! !! +?! ,( ?M 6N!8 J N ( # = ( ?!
4 ! " " (
6> 8 3 > ( 5 ) 5 ( " +! ,( "MB 6 "8 3 ( ( 6 Q??8 K 3 K Q ( ???
! " +? ",( ?M
# &! !( I I (
6G G ?8 G Q G9 ( #& ( & ! +? ?,( MB 6GB8 I G - ( 5 ) I ) ( 7 +?B,( !M" 63 ! 8 # 3 ( ( & ! !! +?! ,( M" 6#!B8 1 #( * C) ( ! +?!B,( B M 6# B8 K # ( # -(
&
& !
% * !+?
B,( ?M
6# 8 K # ( 5 - a * ( &% ! " +? ,( ??MB 6# 8 K # ( E & ( - -
( +? ,( " M B 6E$1!B8 E- ( Q $ C- ( = 1)(
& -( & ! ! +?!B,( ?M?? 6?8 G ( ! ( J N ( $ ( ??
# ( # =
6 8 # A ( 2 ) ( I " +? ,( !M
) I
& !
6 !"8 I ( E ( ! 2 )! ! +?!",( ? M
556
Chapter 6
65 8 5 5 # A ( 1 ( ?
ALGEBRAIC GRAPH THEORY
# ! (4 -!(
65"8 # J 5 5 G ( 1 +$ ,( ?"
# !* (4 -! " !(
65?8 G # J 5 5 ( 1 +$ ,( ?? 6 ?8 1 ( # ( !M
# !8 " !(
! 2
?B +? ?,(
6 B8 1 (
) ( +? B,( ?B M?! 6 8 1 ( 7 (
+? ,( B"M!
6PI"B8 K P 3 E I( ) ( & ! ! !! ! +?"B,( " M" 6P8 E P) (
!! !( # 5 ( ?
61B8 N A 1(
& # ( =
618 = 1 ( (
( BB
7 +?,( !M"
61 #!8 1 1 3 # #& (
) ( & B +?!,( " M"? 61 !B8 N = 1 ( -( & ! ! !! ! +?!B,( M!
Section 6.5
6.5
557
Spectral Graph Theory
SPECTRAL GRAPH THEORY
%& & A # & 1 - 5
3 ( 5 ( > A N 5 I C 3
Introduction 5 + , + ) ,
A
& ) B 9 & )9 ) 3 I C ' 5 C 6I 5 !8 ? ! 5 )9 ( )
6.5.1 Basic Matrix Properties # & - & ) 6# # 8 6 B8 DEFINITIONS
2 ) + , : & 0 + 0 , 0 4
! B
9
2
9 &
2 2
+C
0,
6 )
2 7 (
558
Chapter 6
ALGEBRAIC GRAPH THEORY
2 6 0 4 6 FACTS
2 ) 9 &
( . ( ( : & 9 + =
& ,
2 2 2
) :
9 & 0 ) C( ( :
& @ + , @ 0@ 4 @ 0@ ½ &
2 9 & ( ( ( B( 9 & B
2 ( 9 ) - 9 &
2 $ ( 6 6 +5 9 & ) ( - 7 ) M 6# # 8,
2 $ ( $ + : 7 ) , 5 9 &
( )
: ) )
2 3 ) )
2
1 ) )
) +
7 ?,
2 6
)
B( (
6
REMARK
2
A 7 (
) ) (
EXAMPLES
& + (
- ,
2 2
+ ,½ ½ * ½ ) 2 , B
2
*,
Section 6.5
2 2
559
Spectral Graph Theory
2
+'#) H ,½ ' 4
2
-
2 +'#),½ ' 4 J (
(
2 I+, 5 2 + ,½ B ½
2 2
)
2
1
+ ',+, ' 4 B
+ H , 9 * & 5 2 +'#),½ ' 4 H
2
2
+ + 4 I+,,2 B * * 2 B * *
2 )
1
2
6.5.2 Walks and the Spectrum Walks and the Coefficients of the Characteristic Polynomial DEFINITIONS
2 )
2
& 9 %
FACTS
2
2
$ '
$ 0 9 & ( ' &
2 2 2 2
) -
9 & ( & 0
0
) -
0
) -
0
0
0
'
)
& )
65 8 $ H H H H
( 4 4 B ( ) ) + &
+C 0,,
560
Chapter 6
2
ALGEBRAIC GRAPH THEORY
65 8 $ H ½ ½ H H ½ H
( 4 +, ( ) % ' ( +4 , ) 4 ( +4 , ) 4 REMARK
2
7 " Æ! $
6IN 5? 8 $ & 7 ! Æ
Æ )
) & ' %& + , 7 ( + , + , (
Æ ) % ' 7 ( ) % ( )
+ - ( - , )
Walks and the Minimal Polynomial DEFINITIONS
2
>+, ( >+0 , 4 B
2 * + ) + , H FACTS
2
,+, 4 -
+ 6 ,
2 $ 9 & 0( 0 4 B ' + $( 0 4 B
2
$ , ( , H ) ) $ 7 B
2 2
+ , &
2
) +
7 ( ) \ - \ : ) , EXAMPLES
2
J & ( ( ( I+,( - ( ( 1 )
2
& )
Section 6.5
561
Spectral Graph Theory
OPEN PROBLEM
4
I C ) : E (
) Regular Graphs DEFINITION
2 ' (
+ 6 , ) ,
2 2
D
2
0
D
9 & : 0 C = 6 ( 6 0 :
6= 8 7 9 &
) / $
! + ,(
+ ,
9 ( (
)
) ) ( ( 0
2
$
,
D + ,(
)
9 ( )(
) ' ( )(
'
EXAMPLES
2
! + , ) 1
$
2
D + , A+ , $
566
Chapter 6
ALGEBRAIC GRAPH THEORY
Distance-Regular Graphs and the Hoffman Polynomial
+ / , ( = 0
DEFINITIONS
2
F
) % 0 4 C "
0 4 0 9 &( 0
&(
& + *,
F B
2
F
/
& -
+ *,
FACTS ABOUT THE MATRIX 0"
2
( 2
0
7 ( 0 0
0
4 D (
/ 0
&
) )
4
0
0 0
H + 0
,
0
) ' 4 B
2
) 9 & 0
H
2
= 0
2
)
FACTS ABOUT THE PARAMETERS /
2
- -
4
/ - (
) )
)
2
0 0
- -
0 - ( 0 -
2
-
2
(
Strongly Regular Graphs DEFINITION
2
+ 6 , ) ( ( 6 4 / 4
/
(
EXAMPLES
2
2 A+ , ++
2
3 2 A+
,) ,
, +
,
Section 6.5
567
Spectral Graph Theory
2
2 - +/ , % % 7+/ ,
9 ( % ( 0 : +
)
/
+/ +/
,) +/
,) +/
) , ,),
FACTS
- &
) & 0
&(
) ( 9 &(
2 +
6
7 + 6 ,( +
, 4
6,
¾
2
5 0( 0 ( C (
+ : ( 9( 9 , * C = 0
$
( + 6 ,(
¾
0
2
H +
6,0 H +
,C 4 D
+ 6 ,
2
¾
H +
6, H +
,
$ ( ( ( 0 B
2
7 + 6 ,(
6½ 4 ( 6¾ 4 +6
,) H F
½¾
6¿ 4 +6
,)
F
½¾(
¾
¾
F 4
6 H 6
6 ( ,¾ ( ,¿ ,¾ H ,¿ 4
,¾ 6¾ H ,¿ 6¿ 4
2
$ F 7 ! : ( ,¾ 4 ,¿ 5
2
E
5 6¾ 4 6¿ ( ,¾ ,¿ :
+ 6 ,
)
,¾ ,¿
EXAMPLE
2
¾
7 (
½¾ + /
¾,(
H +
,¾ 4 ,¿ 4 +/
6, H + , 4 ¾ H +/ ,)
H ,)(
6¾ 6¿ 4
FURTHER READING 7 ) &
6 B8
6A I J"?8
568
Chapter 6
ALGEBRAIC GRAPH THEORY
6.5.5 Spectral Characterization E : - 2 C ) Y 7 )
EXAMPLES
2 7 (
½ B & ) : + - ) , )
5
( : )
)
Figure 6.5.3
** *
2 7
) + ,+ H ,+ H ,
Figure 6.5.4
** " *
2 7 ) )
+ , B
+ H ,+ ,
Figure 6.5.5 1 * - $ 2
Section 6.5
569
Spectral Graph Theory
2
7 ½ 5 ( ( % $ ( 9 A+ , + , A ( + - " , ) a
b
c
d
a
e
e
f
f
g
g
a
b
c
d
a
0 *% * *
Figure 6.5.6
Eigenvalues and Graph Operations
E ) DEFINITIONS
2 ( ) ( & ) + , & & 9 ( ( 9
2
+ 9 , ( ( ) ( ( ) & 9
FACTS
2
&
2 6
) ) H 6( 6 6
2
3 ) 7 $ 4 B ,( % ! ) - I ) ) ) ) ) $ , $ !
2
65!8 3
+ , )
-#
+ , 4 - ½ +,- ¾ ¾ +, H - ½ ½ +,- ¾ +, - ½
- ½ ¾
2
!
½ - ¾
¾
65!8 ) ( ))
570
Chapter 6
ALGEBRAIC GRAPH THEORY
EXAMPLES % & ) K = 0
+ K = 0 6# !8, 3 ½ )
2
N% !
) - $ ,
( $ 4 B , !
$ -
( (
2
=
! + ,
) )
( )
REMARK
2
) C J>5 ) N I- S
6IN 5? 8
6.5.6 The Laplacian 3 9 & ) 9
$ ( (
- )
3 (
- )
DEFINITIONS
2
-
: &
&< 0
$ ( A 4
0
9 B 0(
&
9 &
$ ( 4 C ( 0 A ) )
9 &
3
2
67 !8
6
6
6
3 A
% ) 6
FACTS ) 3 ) 3 ;
4 ; + , ) )
) ) * A
3 A
) &
(
9 A )
9 A
+A ,
2
; + , 4 +
2
9+A, 4 ; D
2
B A )
,
2
6
4 ; + ,
Section 6.5
571
Spectral Graph Theory
2 5 6¾ 4 6¿ 4 4 6 4 ( ; + , 4 ¾ 2 ) 2
+
6¾ ½
) ¾, 4
6¾+ ½ , 6¾+ ¾ ,
2 6JBB8 $ (
½ + ,
6¾
EXAMPLES
2
6¾ -
2
6¾
2 2
6¾
2
6¾
+
, 4 + +#),,
+
, 4 + +#),,
6¾ 1
+
, 4
+
, 4
+ , 4
,
FURTHER READING
3 ) A # 6# ?8 & ) # J 6JBB8
References 6A I J"?8 > A ( # I ( J ( )! 5 P ( ?"? 6I"?8 1 I (
%" # !(
# ! 2 ( K 1 D 5 ( ?"?
6I 5 !8 3 I C ' 5 C( 5 - ( $ " +? !,( !M
6IN "8 N I- S(* # N )( $ ( E 1 5 N J >& + H ,½¾( ! +?",( M? 6IN5"8 N I- S( # N )( 5 5
S( C ( +?",( " M?? 6IN 5? 8 N I- S( # N )( = 5( A ( ??
#
2 # !( K )
6I5 ? 8 N I- S 5 5
S( E * & + ,)( )! +?? ,( M! 6N? 8 > N ( ( " +?? ,( ?M
" !
6N=B8 > N 1 = = ( 1
) Y( " &! (
572
Chapter 6
ALGEBRAIC GRAPH THEORY
6N !B8 # N )( ( 2 *
* ( +?!B,( M 67 !8 # 7 ( ) ( ?"MB 6 B8 7 (
'! - +?!,(
2 !* P $( $$( I ( ?
B
6I55! 8 K # ( I ( K 5 ( > 5 ( 3 ( ( " +?! ,( B M! 6 ?8 I ( "
!( I =
6 B8 I ( "
( ??
" ( 5 P ( BB
6=? 8 1 = ( $ ( " ! # +?? ,( ?M 6= 8 K = 0 ( E
( BM
+?
6= ! 8 K = 0 ( > ( M ! &&( N 7 - ( >( # +?!B,
,(
# *
63 ?8 K 3 ( " ! 2 " !( N ( ' # )( ?? 6# # 8 = # # # ( 2 5 ( 1) D 5
( ? 6# ?8 A # ( 3 \ ( !M"
5 &1 !(
)! +??,(
6# !8 # C(
( +?!,( !!M? 6JBB8 # J ( # )( BBB
! 2 " !* # ( '
6 B8 # S O 9 S( 5 ( ' G 9( BB
! " !( 7
65 8 = 5( AC C G -
( ) +? ,( ?M 65!8 K 5-( !
! ! ( J N +>2 7 = ,(
( ?!( ! MB! 65 ? 8 * 5 5
S( 5 + ,)( " +?? ,( ?M!
Section 6.5
573
Spectral Graph Theory
65 !B8 K = 5
( 5 ( BMB ! !( >2 ( = = ( J 5 ( K 5L ( A +?!B, 61 J? 8 E M" +?? ,( !!M ?
*
(
" !
574
6.6
Chapter 6
ALGEBRAIC GRAPH THEORY
MATROIDAL METHODS IN GRAPH THEORY ' $( ) # 2 A N% >& & 5
N # ' $ I : 7 E ! I # " ' # ? >& # I C B 1 ( 1 ( 5
) I # ' # I
Introduction > (
: ( E ( C
- ) 1 +?!MBB, 6!?82 .$ ) ) & )) & % )
/ )
6.6.1 Matroids: Basic Definitions and Examples
&
= 1 +?B!M?"?,
% ? 61 8 DEFINITIONS
2 9 % +9 , + 9 , +9 , ) ) ( ½ ¾ ) +9 , ½ ¾( ) ¿ +9 , ¿ + ½ ¾ ,
7 : ( +9 , +9 , ))
Section 6.6
575
Matroidal Methods in Graph Theory
2 ) ) (
2 2
+ , &
9½
G +9½ , +9¾ ,
G+ , 9¾
( 9½ 4 9¾(
9½
9¾
) -+9 ,(
9
) ,+9 ,
REMARK
2
$ % ) EXAMPLES
2
0 & )
$* 3* " # E$N 9
+ ,( % * "
9
E'JN 5> +9 ,
I$ I'$5 +9 ,
$JN>>JN>J 5>5( ,+9 ,
A5>5 -+9 ,
C + , 2 C
7 2
) 0
C 2 C )
&
2 4 , H
C 2 C ,
4 2 4 4 ,
+ ,(
2 " 6 8(
9 0
& 0 % 7
) "( @
+B , ,
2
3 9 ) +9 , 4 +9 , 4 9 4 9 + ½ , 4 9 + ¾,( ½ ¾ 7 -+9 , 4 ( 9 4 9 608( 0 &
B B B
B B
B B
B B
B B
576
Chapter 6
ALGEBRAIC GRAPH THEORY
1 1 2
3
4
2 G
3
1
5
4 G 2
6 5
6
Figure 6.6.1 9 ½ " %*" " 9
2 $ ) ( % 9 % % $* / * " I
I
9 4 9 + ,
)
9 4 9 608 & 0 %
)
) 7 +,( %
) 7 +,
) %
FACTS
$ ( 9
2 : @ 2 $ 9 ( 9 4 9 + , 2 +1 * $ 618,
) ) ) : 2 + , & < +
, + ,< +
, ) ) 9 ) ( ) ( (
:
2 :
) 2 $ 9 ( 9 2 9 9 ) ) ) ( & )
B
Section 6.6
577
Matroidal Methods in Graph Theory
6.6.2 Alternative Axiom Systems # ) C ) 0 & & E ) ( & ( 6E&?8 (
% $ )
"" 3 ) , $
: ; ,< : ; ) ) , , +, ,< : ; , (
,
&
@
@
)
@
@ 7 7
I
>&
@ @ 7 7
@ 7 7 9
+ , 9 + ,
+
,
+
@ 7 7 9 9
,
2
)
CONJECTURES
9 )
588
Chapter 6
ALGEBRAIC GRAPH THEORY
1! + * I 9 6 !8, 7 % % ( % &
) 1! 7 % % ( /
) ( % &
/ REMARKS
2
7 " )) (
Æ 7 ? 7 " & 9 ) 7 " ) &)
2
7 ) ( > ( &
7 +>, ) % ( ( G 6GBB8 7 +,( ) & &
* I 9 > & 5 I 9 I ) ) 1 61B8 ) ( ( 1 61B8
6.6.10 Wheels, Whirls, and the Splitter Theorem 6 8 % ) ) ) - 5): ( C 6 8 5 ( C ( ) 5 65"B8 ( ( ) J
6J"8 DEFINITIONS
2 7 ( 0 ) & 9 ) + , &
2
7 ( 0 0 0 & ( -
2 3 9
B
B
)
9
B
9
2
/ - ½
/ ( ' ( ' - ' H
EXAMPLES
2
7 0 ( ( 9 +0 , 0 9 +0 , 0
Section 6.6
589
Matroidal Methods in Graph Theory 1 5
1
6
1
(a)
(b)
6
5
(c)
6
5
2
3
2
4
Figure 6.6.5 :;
3
2
3
4
0
4
" :$; 9 +0 , " : ; 0
2 A 7 7 & - 2 : - ) - & 6C C 8 +, C & ) ) C
& C (
7
FACTS
2 6 8 3 : 2
)
+ , 7 ( +
,
) )
-
2 6 8 3 9 ) : 2
+ , 7 +
,
9
9 9)
9 +0 , 0
9
(
2 + 5 65"B8, 3 9 B )
B
9 ( +B , ( 9 5 B 4 9 +0 ,( 4 0 ( 9 0
< B 9 9 +0 ,
: 9 9 9
9 4 9 < 9 4 B < ( ( 9
9
) 5 ( I 6I " 8 + 6I E&?8,( 5 * 5 ) $ - 7 ) ( % 5
2 61 B8 3
7
!
)
) 4 !( ! #L
Figure 6.6.6
# .' $ *""7
!
590
Chapter 6
2 6=8 3 ) 4
ALGEBRAIC GRAPH THEORY
2 65"B8 3 9 ) ) 9 9 4 7
2
3 4 0
2
)
6E&"?8 3
9
7
0
)
0
+ ,
7 !< +
, ' ( ) ) 9 & )
Figure 6.6.7
- #-*
) 6?8 ( ( (
2
6E E&?8 7
B
2
( B
0
6N E E&P?!8 7 ( B B
@
@
+
9
, 9 + , 9 +0 , 0
-
6.6.11 Removable Circuits # ' ) '
C ' 4 = ( # * DEFINITIONS
2 ' & )
) '
'
2 7 ' (
' 9 &
9
Section 6.6
Matroidal Methods in Graph Theory
591
FACT
2 +# , 6#!8 $ '
' H ( ) # * + , ½ +' H , + , &
) )
& )
2 63E&??8 3 9 ) ) 9 $ +9 , +9 , H +9 ,( 9 9 9 +9 , 4 +9 , $ ( + , 4 +9 , +9 , +9 , H ( 9 ) 2 63E&??8 3 9 ) ) 9 $
+9 ,
"
+9 , H +9 , 4 +9 , H ( +9 , H +9 ,
9 9 9 +9 , 4 +9 , & ) )
2 3 ) ) $ + ,
+ , + ,( )
$ (
+ , + ,( ) 2 3 ) ) 5
+ ,
"
+ ,
( + , + ,
2 6 #?!8 3 )
$
( 9 )
2 6#BY8 3 ) $
( ) ) 7 ( # * +7 B, ) ) )
2 65 ?"8 3 )
%
) >& >? ) C & C ) & )
2 6 K??8 3 9 ) ) % $ 9
) 7 7 ( 9 ) +9 , 4 +9 ,
592
Chapter 6
ALGEBRAIC GRAPH THEORY
REMARKS
2 7 ( )
)
2 7 ) ) ( (
( ) ) ( ( (
2 = )) ) ) - >
) EXAMPLES
2 63E&??8 I 2 )
& ! " ? B< ) ! " ? B( & E& 9 ) ) C 9 + , C A 9 + , ) ) ) C 7 B
) ' 4
2 K- 6K"B8 ( ( ) + 6K"B8, = ))* : + ?, ) % 7 "+, 2 6 #?!8 7 ) ( 7 "+),
) ) ) 9 ( # 7
2 7 ( @ ( H ( )
(a) G 1
(b) G 2
Figure 6.6.8 8
+
9
,
9
+ ,
2$*
PROBLEMS
4:;! 6 K??8 $ $ ) $ ) Y 4:;! 6 K??8 $ 9 ) (
9
) Y
Section 6.6
593
Matroidal Methods in Graph Theory
6.6.12 Minimally '-Connected Graphs and Matroids 7 ' ( ' ' ' 7 ' ( DEFINITIONS
2 7 ' ( ' '
'
2 7 ' ( ' 9 '
2 3
9
9
)
9
9
'
EXAMPLES
2
$ , ' (
' 7 ( - 0
0
2
7 7 )
FACTS FOR ARBITRARY CONNECTIVITY
2 6#!8 7 & '
'
(
'
2
6#!?8 7 ' ( )
'
'
'
'
+' , + , H ' '
2
6E&")8 7 ' ( ) + , + , H ' REMARKS
!
7 !B ' 4 ) N 6N !8
6 "8( ( ' 4 ) = 6= ?8 7 ! )
!
) 7 !( ) 7 ? )
( : ) 7 !B
!
6#? 8 ) - ) 9 &
)
& 7 ? ) ) '
594
Chapter 6
ALGEBRAIC GRAPH THEORY
FACTS FOR SMALL CONNECTIVITY
2 $ 9 9 4 9 + ,
( ) +9 , 9 & 2 6N !( "( = ?8 7 ' (
'
'
2 6E&"( E&")8 7 ' ( + , +
,
9
2 61?"8 3
9
9
2 6= ?8 3
+9 , H + '
)
)
+ , +
,
)
'
' < , ' '
9
9
A &)( G - * 1 * (
+?!!,( M
6A !?8 > A &)( E * C (
+?!?,( !MB
6A E&?8 A - K E& (
( M + J 1 ,( I ) ' ( ??
6I " 8 I I ( .# # 9 A # /( N ( J ' ( ?" 6I E&?8 I I K E& ( >& * 1 1 ( +??,( BMB 6I"8 1 = I ( E ( +?!?,( ?M??
B
6N E E&P?!8 N ( A E - ( K E& ( N P ( ' )
( ! +??!,( M?? 6> 8 K > ( 3 * J 1 ( %! ( ! ?A +? ,( !M! 6> ?8 > U ( E & ( " B +? ?,( !M
61B8 K 1 ( A * 9 ( " +BB,( MB
596
Chapter 6
ALGEBRAIC GRAPH THEORY
6GBB8 K 7 ( # = ( G ( &
7 +,M ) (
!? +BBB,( !M?? 61B8 K ( # = ( 1 ( A : ( " +BB,( " !
6 K??8 3 A K- ( ) ) ( +???,( ?M
6 #?!8 3 ( K = ( 5 # ( )
( ! +??!,( BM 6?8 A 5 ) (
( )! ?M! 6=8 N 1 = (
- ( M!
B +??,(
? +?,(
6= ?8 = ( ' L)
( " +? ?,( ! M""
C
L
6= ?8 = ( O C
L ( $ " +? ?,( M 6$1"8 $- 3 1 ) ( 1 ( )! ! +?",( M
& "
6K"B8 A K- ( )
( 9 +?"B,( " M? 6GB8 G G - ( 5 ) d ) ( 7 +?B,( !M" 63 "8 3C ( ) ( +? ",( M 63 8 3 ( 5 ( +? ,( "!M! 63!?8 3 ( E : ( BM!
& !
" " !
+?!?,(
63E&B8 # 3 K E& ( ) C ( ! +BB,( B?MB 6#!8 1 # ( >-
( +?!,( ?M
C
L
6#!8 1 # ( G C 1 ( " +?!,( "!MB
$
6#!?8 1 # ( I % ( M? ! ! + A A )S,( I ) ' ( ?!?
Section 6.6
597
Matroidal Methods in Graph Theory
6#? 8 1 # ( E
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