--Jonathan Gross and Jay Yellen

• ADDITIONS in 2nd PRINTING (July 1999)
• ERRATA IN 1st PRINTING CORRECTED 
  IN 2nd PRINTING (July 1999)
• ERRATA IN 2nd PRINTING to be CORRECTED 
  IN 3rd PRINTING

ADDITIONS in 2nd PRINTING (July 1999)

Copyright page: now identifies the graph on the cover as the wrapped-butterfly interconnection network.

Notes added to Preface for the second printing:

1. An Instructor's Solutions Manual for the exercises in this text is available upon request from the publisher. The website for CRC Press is www.crcpress.com.
2. The website for the authors is www.graphtheory.com. Suggestions from readers are welcomed.
3. We thank our students for locating various minor errors in the original printing, which have been corrected in the second printing.

ERRATA IN 1st PRINTING CORRECTED IN 2nd PRINTING (July 1999)

9 Exc 1.1.9: the self-loop at vertex u gets label c.

15 Exc 1.2.11 revised

a. Draw all the 3-vertex tournaments whose vertices are u,v,x.

b. Determine the number of 4-vertex tournaments whose vertices are u, v, x, y.

41 ln 2:

• Irish mathematician Hamilton

54 exercise references

• in Exer 2.1.18 graph of Exercise 2.1.14
• in Exer 2.1.19 graph of Exercise 2.1.15
• in Exer 2.1.20 graph of Exercise 2.1.16
• in Exer 2.1.21 graph of Exercise 2.1.17

68 Exer 2.3.32:

• graph on right now has two edges between vertices r and t

114 below subhead, in 2nd sentence

• search is performed until it terminates at a leaf w

114 recursive Algorithm 3.6.3 now has two parameters T and x

142 pf of Prop 4.4.1 2nd paragraph:

• ln 1: true for all LABELED trees
• throughout paragraph: replace k's by n's

143 Algorithm 4.4.2, 2nd line from bottom of box:

• change "in list P" to "in list L"

165 sentence above "REVIEW FROM":

• two brace-pairs changed to angle brackets.

180 exc 5.1.20:

• no 3-connected SIMPLE graph ...

183 proof of Prop. 5.2.5, last sentence:

• follows by Corollary 5.1.6.

196 Example 5.3.4:

• replace "size 3" and "both 3" by "size 4" and "both 4"

196 ln -4 instructions for exc 5.3.9 and 5.3.10:

• u-v paths (not s-t paths)

200 Exercise 5.4.15: Prove Corollary 5.4.3

200 Exercise 5.4.16: Prove Corollary 5.4.4

200 Exercise 5.4.17: Prove Corollary 5.4.5

206,207 - Instructions for exc 6-6 thru 6-9, now say

• "Begin the construction at vertex s."

206 figures for exc 6.1.8 and 6.1.9 inserted

206 - figures for exc 6.1.6 through 6.1.9:

• leftmost lower vertex is now labeled "s"

207 - exc 6.1.13: new figure

210 Example 6.2.4: ln 1

• "the second property" --> "Proposition 6.2.2"

219 exc 6.2.5: "Theorem 6.2.1" --> "Proposition 6.2.1"

220 exc 6.2.19: change UC-fragment list to:

• GGAAAG, GU, C, C, C, C

221 - exercises 6.2.25 through 28:

• ALL of these matrices now have f_ii=0

226 - Example 6.4.1: in Gray code

• delete the second occurrence of 000

229 Algo 6.4.1:

• 3rd instruction should be "initialize i=0"
• insert "i := i+1" as first line in body of while-loop

231 Algo 6.4.3: lines 2 through 4 in body of algo now read:

• Let O be the subgraph of G ...
• Find a minimum-weight perfect matching in O.
• Create an eulerian graph H by adding the edges in M* to T*.

254 Fig 7.3.5 caption now reads

• f:K_3,3 --> K_2,3

259 Exer 7.3.21:

• ... equal to 4, first show that 4 is ...

286 ln 1:

• the graph of Figure 8.3.7

331 ``Corollary'' 10.1.15

• renamed Proposition 10.1.15.

332 Example 10.1.9:

• numerator should be 7 instead of 8; and
• first line below display: 4-colorable --> 4-critical

332 Prop 10.1.17 first ln of pf:

• delete ``then''.

343 pf of Th10.2.2, 1st line of 2nd paragraph:

• If all five colors were not ... --> If not all five ...

353 Example 10.3.5:

• 3-coloring --> 4-coloring

369 exc 11.1.25:

• Then digraph D has at least k+1 ...

380 exc 11.2.24:

• ... of Application 11.2.4....

389 exc 11.3.1: changed to

• "Group the 3-vertex tournaments into isomorphism classes."

394 Algorithm 11.4.2: move increment for counter k

• to first statement under "if low(t)=dfnumber(t)"

395 - new figures for exc. 11.1.1 through 11.4.4 now have cross arcs

395 - exc 11.4.2:

• the unlabeled vertex should be labeled "f"

402 ln 1:

• subscript of V should be N

405 Exer 12.1.1, 12.1.2 figures inserted

424 exc 12.3.18:

• ... is a collection of r arc-disjoint ...

427 last line above Algo 12.4.1:

• edge x in graph G.

431 Fig 12.4.8: change of bold vertices

445 immed above Fig 13.1.5

• ...|Aut(K_n,n)| = 2(n!)²

458 Lemma 13.4.2:

• Stab(y) --> |Stab(y)| in several spots

484 Exer 14.1.20: 2n+1 --> 2n-1

491 Fig 14.2.13: change voltage on edge c from 000 to 001

494 Fig 14.3.1: place voltage (1 2 3) on edge b

499 ln 3:

• proofs of Theorems 14.4.1 through 14.4.4 ...

502 ln 4:

• ... C(Z_2n, {1,n}), while adopting ...

505 ln 17: second equality:

• the fraction (n-2)/2 replaces (n-1)/2

515 Fig 15.2.4 corrected figure

520 Prop 5.3.1:

• Let h:G --> S be an oriented ...

532 TERMINOLOGY:

• A cyclic permutation of the constituents ...

549 def. of identity element:

• a*e = e*a = a

65 Theorem 2.3.3:

• The formula for the number of edges of G is correct only for graphs with no self-loops. To get the correct formula for graphs containing self-loops, count only proper edges in the given formula and add (1/n-1)(sum of the number of self-loops appearing in the vertex-deletion subgraph list).

79 Exercise 2.5.2:

• label the self-loop at u with a "j".

90 Exercise 3.1.2:

• should be "12-vertex" and "10 edges".

109 Line just above 3. In-Order Traversal:

• (see Exercise 3.5.3) should be (see Exercise 3.5.10).

109 3. In-Order Traversal:

• The geometric description is not correct. To make the description correct you must treat any vertex with only one child as if it had two children, representing the missing child with an edge without an endpoint (a null). The left-walk goes around these special edges but finds no vertex to list.

112 below Example 3.6.1

• proceed iteratively to the right (not to the right-child) as long as possible.

112 ln 1

• A binary-search tree (BST) is an ordered binary tree ...

113 Algo 3.6.2

• name BSTsearch(T,t) must match recursive call.

115 Exercise 3.6.8

• Replace "insert 35" with "insert 33".

136 Example 4.3.1, 4th line from bottom

• Replace "Observe that the cut-vertex w" with "Observe that the non-cut-vertex w".

136 Example 4.3.1, next to last sentence.

• Replace sentence with "Also notice that when either of the cut-vertices v or x is not the root, it has a child none of whose descendants is joined to any of the proper ancestors of the cut-vertex. "

137 Proposition 4.3.3, last line of statement.

• Replace "an ancestor of v" with "a proper ancestor of v ".

172 exc 4.8.8:

• ... mutually ADJACENT EDGES ...

177 7th line from bottom

• Replace the last "y" in this line with an "x".

177 3rd line from bottom.

• Replace "vertex x" with "vertex y".

205 computational note:

• remains in the stack (not queue)

275 Cor 8.1.2: A path from one point on the boundary of a disk through the interior to another point on the boundary separates the disk.

344 - remark above Exercises:

• Robertson, Sanders, Seymour, and Thomas [RoSaSeTh97]

365 TERMINOLOGY

• dag is acronym, not "mnemonic"

417 proof of Prop 12.3.2 ||2, ln2

• "premise", not "condition"

428 REMARK ln 3: reference should be:

• Edmonds: ``Paths, Trees, and Flowers'' Canad. J. Math. (17), 449-467, 1965

461 pf of Prop 13.5.1:

• delete the last two sentences of the proof

503 Example 14.4.6

• ln 3: Z_n --> Z₃ (twice)
• ln 4: ... Figure 14.4.9

503 Example 14.4.6 ln 4:

• ... Figure 14.4.9

565 - Th89 is cited on p528 (not on p513)