# Combinatorial Mathematics - Fall 2003 Typos

This page lists the typographical errors that have been discovered in the Fall 2003 pre-publication version of Combinatorial Mathematics, by Douglas B. West. The text went to the copy shop on September 15; the corrections listed were discovered since that printing. This page is of interest only to those persons having a copy of this draft, particularly the students in my course, other users of the text, and reviewers. Please send comments and corrections on the book to west @ math.uiuc.edu. Contributors are noted in parentheses; "(JG)" denotes John Ganci.

## Category 1: Mathematical typos/corrections to text

• p36 Exr1.2.29: The given identity is nonsense. It should be \sum{m choose k}{n+k choose m}xk = \sum{m choose k}{n choose k}xm-k(1+x)k
• p94 Exr2.3.12: The "j" in the denominator should be "k", and the final "=n" should be "=n-1"
• p101 Exm3.1.12: The index on the inner sum in the first double summation should be k, not n (Sujay Sanghavi)
• p113 top line: the exponent on (1-x) in the denominator should be 2, not -2 (Mike Rosulek)
• p122 Exr3.2.38: The sum as stated in part (a) is unbounded. The bottom term in the first binomial coefficient should be 2k rather than k.
• p145 Prop3.4.4: "an,k" should be "an,3" twice (JG)
• p146 Rem3.4.6: the "-2\sqrt n" should be in the exponent
• p174 second paragraph: "cs" should be "es" (Mike Pelsmajer)
• p184 Exr4.1.52: "at a time" should be "at a time upward or rightward"
• p187 Def4.2.4: "\pi x" should be "\pi(x)" (JG)
• p189 bottom: "acts a set" should be "acts on a set" (JG)
• p246 Thm5.2.8: second sentence of second paragraph should start with "Let" (JG)
• p251 Exr5.2.10c "\eps+a/2" should be "\eps" plus half the average degree of G (Jennifer Vandenbussche)
• p254 Def5.3.1: in the second paragraph, "x,y" should be "u,v" (JG)
• p263 Exr5.3.14: as introduced in Def5.1.3, the cycle should be specified as "[v1,...,vk]" not "(v1,...,vk)" (JG)
• p306 Alg6.3.10: the description should start by assuming a weighted cover (u,v) (Michael Pelsmajer)
• p317 Lem7.1.4: "in the k-expansion" should be "in the k-split" (Michael Pelsmajer)
• p332 Def7.2.15: the definition also should specify that the endpoints in U are distinct (Michael Pelsmajer)
• p332 Thm7.2.17: "x,V(C) fan" should be "x,V(C)-fan" (JG)
• p340 Lem7.2.29: "u,v shortcut" should be "u,v-shortcut" (JG)
• p348 Thm7.3.2: "set S\esub V" should be "nonempty set S\esub V(G)" (Michael Pelsmajer)
• p348-9 the notation c(H) for the number of components of H should instead be #(H), since c(H) is used later in this section for the circumference of H
• p365 Exm8.1.4: r must be at least 2 (JG)
• p373 Exr8.1.17: should require that G be connected (or deal separately with spanning trees in each component)
• p380 Thm8.2.10: the last instance of "d(v)" refers to the degree of v in the subgraph induced by V(C) (Michael Pelsmajer)
• p388 Exr8.2.29: the bar over K4 in the last line should extend over the rest of the expression (Noah Prince)
• p393 intro to Vizing's Theorem: add left brace on maximization in definition of \mu(v) (JG)
• p393 Thm8.3.8: in the statement, "graph" should be "multigraph" (Michael Pelsmajer)
• p433 Thm9.3.10: in the statement, "planar" should be "plane"
• p441 Exr9.3.21: "\in G1" and "\in G2" should be "\in V(G1)" and "\in V(G2)" (JG)
• p448 Exm10.1.6: last line should be "{m\in [2n]: m=(2k-1)2j-1 and j\in\NN}" (JG)
• p459 Exr10.1.10: "log2(\chi(G))" should be "\ceiling(log2(\chi(G)))" (Michael Pelsmajer)
• p460 Exr10.1.12: "(mod n)_i" should be "(mod n_i)" (JG)
• p461 Exr10.1.23a: since the center of a graph is defined to be the set of vertices of minimum eccentricity, the statement should be that the center of a tree consists of one vertex or two adjacent vertices
• p461 Exr10.1.26: "longer more" should be "more" (JG)
• p467 before Lem10.2.6 "proves in the worst case" should be "probes in the worst case"
• p467 Lem10.2.6: add the statement the hypothesis that M has size at least 2 to avoid a degeneracy when n=1. Note the case n=1 when starting the proof. The main argument can be simplified as follows: If there is a consistent set of size at least 2n-1, then any 2n-1 elements of it form a consistent set of size 2n-1. Hence we may assume that P has size 2n-1, and then the "central set" reduces to a single element. (Michael Pelsmajer)
• p468 Lem10.2.6: in line 3 of the last paragraph, the subscript "\ceiling{n/2}+1" should be "\floor{n/2}+n". (Michael Pelsmajer)
• p489 Lem10.3.18: "sum at most [m]" should be "sum at most m" (JG)
• p490 Def10.3.20: "B0,B1" should be "B0,B1 of [m]", and it should be noted that B0 may be empty (JG)
• p491 Def10.3.21: "layered" should be "layered subspace" (JG)
• p495 Exr10.3.10: Both instances of "W(" should be "w(" (JG)
• p499 Prop11.1.2: "Kn,r" should be "Kr+1", and add mention afterward that e(Tn,r) is asymptotic to (1-1/r)n2/2 for fixed r (Michael Pelsmajer)
• p501 Lem11.1.7b: "(n-2)e" should be "(n-2)m" (Michael Pelsmajer)
• p502 top: in line 4, the expected number should be "{n\choose r}2-{r\choose 2}" (Michael Pelsmajer)
• p511 Exr11.1.19: last "F" should be "|F|" (JG)
• p512 Exr11.1.31: "ex*(D : H)" should be "ex*(D;H)" (JG)
• p534 Thm11.3.12 middle: "W=w1, . . ., wt" should be "W={w1, . . ., wt}" (JG)
• p540 before Def12.1.6: "A chain" should be "A finite chain" (Michael Pelsmajer)
• p540 Exm12.1.7: as defined, the pairs in the cover relation should be {(a,c), (b,c), (c,d))} (JG)
• p541 Exm12.1.9: "2n is also called" should be "2n is also called"
• p544 after Def12.1.16: "set brackets" should be "set brackets and commas" (JG)
• p545 Thm12.1.19: the second diagram does not correctly indicate the second case in the proof, because this poset has a maximum antichain that omits both a minimal and a maximal element; the picture will be corrected (Chris Andrews)
• p550 Exr12.1.8: "common poset condition that adjacent vertices must satisfy.)" should be "common", and "a G without" should be "a graph G without" (JG)
• p563 Exr12.2.8b" This is false for a poset whose comparability graph has an edge and an isolated vertex. The problem should require that P is a finite poset whose diagram is connected. (Michael Pelsmajer)
• p564 Exr12.2.14: "hypergraph" should be "hypercube"
• p576 Exr12.3.14: "completing following" should be "completely following" (JG)
• p578 Prop12.4.4: "Liyi" should be "Liyi" (JG)
• p588 Lem12.4.32: in the display, "f(x,x)" should be "f(y,y)"; also "\deltaxy" should be "\deltax,y" (JG). Also, the earlier part of the proof has been rewritten to clarify that we are determining the existence and uniqueness of the solution to the equations required by g being a left inverse of f
• p590 Exm12.4.38: "chain \omega" should be "chain consisting of N under the usual ordering < " (JG)
• p594 Exr12.4.11: "|Bn+1" should be "n+1". Note: Many instances of "n+1" in this chapter should have the "+" in bold, referring to a single chain of size n+1 (not the union of two chains of sizes n and 1) (JG)
• p596 Exr12.4.26: "y\join z" should be "y\join z}" (JG)
• p602 before Thm13.1.10: "2P(L)" should be "2|P(L)|" (JG)
• p619-620 Rem13.2.11 and Lem13.2.13: to make this correct, we must define a normal bipartite poset to be a bipartite poset such that the diagram is connected and each element is incomparable to some element on the other level. The statements are true in this class
• p627 Exr13.2.9: "with height" should be "with even height" (JG)
• p635 Exm13.3.7: "P(x < y < w ||)" should be "P(x < y < w | Q)" (JG)
• p644 Prop14.1.5: equality holds when x=0
• p665 Lem14.2.12: "Vi\cap Vj" should be "Vi\cup Vj" (Apu Kapadia)
• p681 Thm14.3.15: "\alH'" should be "EH'" (Michael Pelsmajer)
• p682 Rem14.3.16(4): "E(XiXj may" should be "E(XiXj) may" (JG)
• p689 Thm14.3.26: four instances of "e" should be "e" (JG)
• p692 Exr14.3.10: "\rho(G)=\maxG\esub Hm(G)/n(G)" should be "\rho(H)=\maxG\esub Hm(G)/n(G)" (Michael Pelsmajer)
• p693 Exr14.3.13: "It is true" should be "Is it true" (JG)
• p693 Exr14.3.14: "Section 4.5" should be "Section 16.2" (JG)
• p698-9 Thm14.4.5 and Thm14.4.7: each has in its proof an "e" that should be an "e"
• p700 Thm14.4.9: "P(X\ge n(p+t)" is missing right parenthesis (JG)
• p702 Thm14.4.15: "ck" should be "ci" (Sujay Sanghavi)
• p708 Exr14.4.6: extra right parenthesis in last expression of part (a), and "pI" should be "\pi" in last expression of part (b) (JG)
• p739 Rem15.3.2: index of summation should be \sigma (JG)
• p752 Thm15.3.31: the summations in the display are over vivj\in E(G), not ij\in E(G) (JG)
• p753 Cor15.3.32, Thm15.3.33: "\partial S" should be "|\partial S|" (\partial S was defined earlier to be N(S)-S; this notation will be used instead here)
• p771 bottom line: "1 (mod p)(x)" should be "1 (mod p(x))" (JG)
• p774 Exm 15.4.41: before the display, "1+x" should be "1+y" (JG)
• p775 Lem15.4.43: the appearances of lowercase "l" in the proof should be "n" (JG)
• p776 Exm15.4.46: in the display, "(mod p)3" should be "(mod p3)" and "(mod p)5" should be "(mod p5)" (JG)
• p781 Exr15.4.7: "(mod p)(x)" should be "(mod p(x))" (JG)
• p781 Exr15.4.9: "multiplication" should be "under multiplication by elements of the ring" (JG)
• p788 figure: in the figure on the right, the labels v1, v2, v3 are spurious and should be deleted
• p787 Thm16.1.10: in the last paragraph, one case is omitted. If ua is in the interior of C, then we must consider the cycles formed by ua with both u,a-paths on C to find the desired uniform angles in addition to a being 1-uniform. (Michael Pelsmajer)
• p793 Exm16.1.25: on the right side of the inequality before the display, "Kn" should be "Kn-1" (JG)
• p799 Exr16.1.15: "cr(K7,7=" should be "cr(K7,7)=" (JG)
• p801 App16.2.3: In the last paragraph, all four occurrences of the subscript "ik" should be ik (JG)
• p819 Thm16.3.20: "logarithm cn2 of 3{tn(n-1)} grows" should be "logarithm of 3{tn(n-1)} grows quadratically with n,"
• p829 Cor17.1.7: it should be specified that each pi is prime (JG)
• p833 Prop17.1.18: "matrix n exists" should be "matrix exists" (Sujay Sanghavi)
• p834 Prop17.1.19: "-1s" should be "-1" (JG)
• p837 Thm17.1.26: "commutativity" should be "associativity"
• p839 Thm17.1.29: in the expression "\alphaiy1", the subscript on y should be i, not 1 (JG)
• p842 Thm17.2.4: second "|B|\ge|A|" should be "|A|\ge|B|" (Sujay Sanghavi)
• p855 four times, and five more on next two pages: "(mod()xv-1)" should be "(mod(xv-1))" (JG)
• p863 Lem17.3.12: "Y a 1-factorization" should be "Y has a decomposition" (JG)
• p865 bottom: "(mod 1)2" should be "(mod 12)" (JG)
• p870 Thm17.3.26: "(n{" should be "(n,{" (JG)
• p873 Thm17.3.34: "other than 2 and 6" should be "outside {1,2,6}"
• p875 and thereafter: the occurrences of underlined "2" in this chapter should be 2 (JG)
• p889 Def18.1.29: "dualmatroid" should be "dual"
• p892 Def18.1.40: "F\esub" should be "F\esubE" (JG)
• p910 Exr18.2.1: "S1,S1" should be "S1,S2" (JG)
• p910 Exr18.2.5: missing left brace on set definition (JG)
• p917 Exm18.3.17: "transversal B" should be "transversal of a family B"
• p925 Lem18.3.31: "obtain a in which" should be "obtain a minor of M in which", and "not spanned by (X)" should be "not spanned by X"
• p926 end of Thm18.3.32: "r'({y}=" should be "r'({y})=" (JG)
• p926 second display: last "1" should be "1T", and in the next paragraph a "c" should be "cT" (Michael Pelsmajer)
• p938 Cor19.1.11: "b > 0" should be "br > 0" (Michael Pelsmajer)
• p940 Thm19.1.14: "before the pivot nonbasic variables" should be "before the pivot by setting the nonbasic variables"
• p944 Exr19.1.21: the figures shown the cartesian second powers; the strong second powers also need the diagonals in each of the small squares. Also "db" should be "db"
• p944 bottom: in the displays, each "1" should be "1" (JG)
• p953 Exr19.1.8: "against and" should be "against T and" (JG)
• p959 Exm19.2.11: the conservation constraint should have f(vy) and f(xv), not f(xy) and f(xv)
• p961 App19.2.13: "T\cap X\cup S\cap Y" should be "(T\cap X)\cup (S\cap Y)" (JG)

## Category 2: Comments, clarifications, and cross-references

• p132 Exm3.3.24: due to a reordering of material, the term "involution" now appears before its definition. This will be fixed.
• p168-9 Thm4.1.22: this application is long and is less compelling than the application to determinants and path systems, so it may be demoted to an exercise
• p171 Thm4.1.26: to avoid confusion with the usage of "intersecting family" of sets in Chapter 11, the usage of "intersecting path system" to mean a system in which some two paths intersect will be eliminated from this proof. Also (Mike Pelsmajer), the digraph need not be acyclic (Mike Pelsmajer)
• p188 2nd parg: a more formal and general definition of orbit will be inserted: When G is a group of permutations of C and v\in C, the orbit of v is {\pi(v): \pi \in G}. (JG)
• p268 Prop5.4.3: the fact that connected graphs have spanning trees is not explicitly stated until Cor5.4.5. Instead of citing this, one can use the following: "Conversely, suppose that G is connected. By Prop5.3.12, every graph with n vertices and n-2 edges is disconnected. Hence every edge of G is a cut-edge, which implies by Prop5.3.11 that every edge lies in no cycle. Therefore, G is acyclic." (JG)
• p292 Thm6.2.6 and elsewhere: In a title like "Tutte 1-Factor Theorem", a notation-prefixed word should have the word part capitalized
• p298 Exr6.2.15: this exercise is moving to Chapter 15, where the material on permanents went (JG)
• p341 Thm7.2.31: last sentence of proof clarified to "After k lifts, also \dlt(\{z\})=k, so now \dlt(X)\ge k whenever \nul\ne X\subset V(G) (JG)
• p342 Lem7.2.32: in the first line of (2), "R" should be "W"
• p346 Exr7.2.39: "d(v)=2" should be "d(v)}=2" (JG)
• p349 Thm7.3.5: before display, parentheses should be brackets in specification of cycle (JG)
• p377 Def8.2.4-Lem8.2.5: this is a bit confused, since "uniquely colorable subgraph" hasn't been defined, and the only such graphs are complete graphs. This lemma will be simplified merely to exclude clique cutsets, and the concept of uniquely colorability will be deleted. (Michael Pelsmajer)
• p391 Def8.3.3: The concept of k-factorization is not used in the text and is being dropped from this definition (Michael Pelsmajer)
• p417 bottom: the missing definition of "tangency graph" has been added (and the grammar of "a tangency graphs" corrected) (JG)
• p419 Def9.2.11 and before: The notation A\Vert B is not used elsewhere and is being deleted from these two places (Michael Pelsmajer)
• p452 Thm10.1.11: the display after the proof illustrate the partition into pairs of elements differing by k; "as illustrated below" has been added in the appropriate place
• p500 before "Counting Cliques": "Chapter 5" should be "Chapter 17" (Noah Prince)
• p522 Thm11.2.21: this proof can be expressed more simply and intuitively in terms of the complement of S. That is, let T be a maximal subset of [n] that is expressible as the union of disjoint members of I. Now H consists of members A in I such that T-A is also in I. The resulting discussion is much cleaner, even though this is the same family H as before. (Michael Pelsmajer)
• p646 after Thm14.1.8: "Corollary 14.1.22" should be "Corollary 14.2.5"
• p653 before Exercises: "Theorem 16.1.22" should be "Theorem 16.1.28", and the statement should say "of a graph with n vertices and m edges has at least m3/(64n2) crossing pairs of edges, when m\ge 4n" (Sujay Sanghavi)
• p790 before Thm16.1.20: "Corollary 13.1.18" should be "Corollary 13.2.18"
• p795 before Thm16.1.31: "??" should be "Exercise 16.1.19"
• p809 Thm16.3.1: "Definition 16.3.39" should be "Definition 16.3.6"
• p822 Exr16.3.3: "??" should be "Theorem 16.3.5"
• p838-9 Due to a rearrangement of material, some statements are motivated using projective planes before projective planes have been defined; this will be changed
• p848 afterThm17.2.14: "Section 3.3" should be "Section 10.1"
• p858 Exr17.2.14: moves to Section 17.3
• p859 second paragraph: "??" should be "Example 11.3.9"
• p866 before Exm17.3.17: "??" should be "Theorem 17.3.15"
• p884 Exm18.1.20: "Exercise 4" should be "Proposition 18.3.23" and "Exercise 5" should be "Example 18.3.30"
• p888 after heading: "Section 2.5" should be "Section 9.1"
• p892 after Thm18.1.34: "Exercises 11-44" should be "Exercises 11 and 44"
• p896 before Thm18.1.47: "Theorem 18.2.12" should be "Theorem 18.2.11" (JG)
• p904 Exm18.2.6: "??" should be "Corollary 6.1.9"
• p914 after Prop18.3.7: "Section 15.1" should be "Theorem 18.1.44"
• p916 after Exm18.3.15: "??" should be "Example 18.1.13"
• p919 bottom: the characterization of regular matroids by representability over all fields will not appear in this text
• p920 after Thm18.3.22: "Section 15.1" should be "Section 9.2". The last sentence of the next paragraph should be deleted, as it refers to items that have been elminated from the text
• p920 Exm18.3.24: "Chapter 19" should be "Chapter 17" (JG)
• p922 after Cor18.3.27: "Chapter 19" should be "Section 15.4" (JG)
• p942 after Thm19.1.18: "Section 4.4" should be "Section 15.2"
• p943 after Thm19.1.20: "Section 6.4" should be "Chapter 18"
• p951 before Def19.1.39: "199?" should be "1991"
• p969 Exm19.2.22: "Section 3.4" should be "Section 12.3"
• p975 after Thm19.2.26: "Section 3.4" should be "Section 12.1"
• p975 Exm19.2.27: "Chapter 16" should be "Corollary 4.3.17"
• p981 Exr19.2.20: The dates for the references are both 1980
• p982 bottom: "Section 14.1" should be "Section 19.1"
• p990 Prop19.3.19: "Chapter 2" should be "Corollary 6.2.12"

## Category 3: Minor typos and corrections

Note: Many minor corrections involving addition, deletion, or alteration of one punctuation mark have been implemented without being listed here. Some corrections to capitalization are also omitted.
• p114 before Thm3.2.19: "Theorem 4.1.21" should be "Theorem 4.1.22" (JG)
• p132 Exm3.2.23: "the sum the Stirling" should be "the sum of the Stirling"
• p143 top: "Example 1.1.21" should be "Corollary 1.1.21" (JG)
• p145 Rem3.4.6: "it is reasonable to argue that this partitions will arise" should be "then the same growth rate should be valid for partitions using `small' parts."
• p148 Prop3.4.11: "is the resulting" should be "is the result" (JG)
• p150 summary: "k-elements multisets" should be "k-element multisets" (JG)
• p152 SID: "Example 1.1.21" should be "Corollary 1.1.21" (JG)
• p155 Exr3.4.19i-20: delete "(*)", replacing "(*)" in 3.4.20 with reference to 3.4.19b. Delete comma after "Andrews" (JG)
• p162 before App4.1.10: "Example 2.3.2" should be "Proposition 2.3.2" (JG)
• p168 before Thm4.1.22: "(barred permutation" should be "(barred permutations" (JG)
• p170 before Def4.1.23: "a informal" should be "an informal" (JG)
• p171 before Thm4.1.26: "we restriction" should be "we restrict" (JG)
• p172 Thm4.1.26: "y\sigma(i)}-" should be "y\sigma(i)-". Also "(the word form) of" should be "(the word form of)" Also "\tau(\tau(P)" should be "\tau(\tau(P))" (JG)
• p173 Prop4.1.27: "submatris" should be "submatrix" (JG)
• p175 Thm4.1.30: "(i,j) entry" should be "(i,j)th entry" (JG)
• p184 Exr4.1.51: in the hint, ". as follows" should be "as follows" (JG)
• p187 before Def4.2.3: "we when" should be "when we" (Sujay Sanghavi)
• p200 Exr4.1.14b: "part (b)" should be "part (a)" (JG)
• p201 Exr4.1.22: "Polya" should be "P�lya" (JG)
• p202 Exr4.1.29: "clases" should be "classes" (JG)
• p202 4.3intro: "combintorial" should be "combinatorial" (JG)
• p216 Thm4.3.22(2): "in row of" should be "in row 1 of" (JG)
• p221 Lem4.3.30: In the second paragraph, after defining B, begin a new sentence with "We have" (JG)
• p221 above Lem4.3.31: "an nondecreasing" should be "a nondecreasing" (JG)
• p247 above Thm5.2.9: "extremal instances" should be "extremal instance" (JG)
• p263 Exr5.3.10: "entries Prove" should be "entries. Prove"
• p264 Exr5.3.18: "multigraph graph" should be "multigraph" (JG)
• p269 top line: "lies is" should be "lies" (JG)
• p284 Thm6.1.8: ", Let" should be ". Let" (JG)
• p292 before Def6.2.5: "sufficiency condition" should be "sufficient condition" (JG)
• p297 before Exercises: "Exercise 44" should be "Exercise 43" (JG)
• p309 Rem6.3.13: "M remain" should be "M remains" (JG)
• p310 before Def6.3.14: "women; we" should be "women, we" (JG)
• p325 section intro: "is a stronger than" should be "implies" (JG)
• p338 Lem7.2.29: "u,v-shorcut" should be "u,v-shortcut" (JG)
• p352 before Thm7.3.12: ", this condition" should be "; the Chvatal-Erdos condition" (JG)
• p356 after Lem7.3.15: "Dirac's theorem" should be "Dirac's Theorem" (JG)
• p360 Exr7.3.20: missing reference is to Exr7.3.19 (which also needs a period before "Prove"). Several subsequent exercises need capitalization on titles ("Condition" and "Theorem") (JG)
• p362 Exr7.3.37: "a n-vertex" should be "an n-vertex" (JG)
• p372 Exr8.1.9: Missing right parenthesis on "(Blatter " (JG)
• p379 Lem8.2.9: "its internal" should be "the internal"; and "adding uv" should be "adding uv"
• p388 Exr8.2.31: "Choose the partition" should be "Choose a partition"
• p389 Exr8.2.33: "tree use k colors" should be "tree uses k colors" (JG)
• p397 Exr8.3.16b: "an proper" should be "a proper" (JG)
• p397 Exr8.3.22b: "an k-regular" should be "a k-regular" (JG)
• p411 Exr9.1.22: "two triangle" should be "two triangles", and the figure showing a flip is being added (JG)
• p412 Exr9.1.36: "assumed that" should be "assuming that" (JG)
• p413 2nd parg: "every planar graphs" should be "every planar graph" (JG)
• p413 Def9.2.1: "then then" should be "then the" (JG)
• p428 1st parg: "we switching" should be "switching" (JG)
• p430 Exm9.3.5: "some case combine" should be "some cases combine" (JG)
• p432 Lem9.3.9: "an vertex" should be "a vertex" (JG)
• p433 Thm9.3.10: "available color" should be "available a color" (JG)
• p435 1st sentence: "refsXhad" should be "Exercise 18" (JG)
• p441 Exr9.3.23: "outside Ct." should be "outside Ct)." (JG)
• p449 Exm10.1.7: "possibile" should be "possible" (JG)
• p451 Exm10.1.10: "77 distinct number" should be "77 distinct numbers" (JG)
• p451 Thm10.1.11: "over all list" should be "of a list" (JG)
• p453 before Def10.1.15: "Exercise 8.1.28" should be "Exercise 8.1.29" (JG)
• p454 before Exm10.1.18: "number, it appeared" should be "number appeared" (JG)
• p455 Exm10.1.18: "Blanches" should be "Blanche" (JG)
• p456 bottom: "procedes" should be "proceeds" (JG)
• p458 bottom: "using in forming" should be "used in forming", and "N-1st" should be "(N-1)th" (JG)
• p471 Rem10.2.12: "Remark 8.1.17" should be "Remark 8.1.18" (JG)
• p473 Thm10.2.15: first paragraph of proof is missing two periods at ends of sentences (JG)
• p476 Exr10.2.26: "monchromatic" should be "monochromatic"
• p477 Exr10.2.32c: "case)." should be "case.)" (JG)
• p479 Thm10.3.1: "Give that" should be "Given that" (JG)
• p498 Exr10.3.15: "xn winning" should be "xn and winning"
• p500 after Thm11.1.5: "Chapter 5" should be "Chapter 17" (JG)
• p502 first paragraph: "Theorem 10.2.12" should be "Theorem 10.2.11" (JG)
• p509 Exr11.1.7: "not r-partite graph" should be "not r-partite" (JG)
• p509 Exr11.1.8: "Given an graph" should be "Given a graph" (JG)
• p510 Exr11.1.17: missing parenthesis at end (JG)
• p511 Exr11.1.18: missing accent on Erdos (JG)
• p521 after Conj11.2.20: missing accent on Chvatal (JG)
• p523 Thm11.2.22: "would is below" should be "is below" (JG)
• p523 Thm11.2.22: "since F*\cup {C} a star" should be "since F*\cup {C} is a star" (JG)
• p525 Exr11.2.16: missing "a" in "Lov�sz"
• p526 Exr 11.2.21: "that is Z" should be "that is, Z" (JG)
• p526 Exr 11.2.24: "includesa" should be "includes a" (JG)
• p546 Thm12.1.20: "n-k chain" should be "n-k chains"
• p572 Rem12.3.15: "ranks sizes" should be "rank sizes" (JG)
• p574 Exr12.3.7: ", Let" should be ", let" (JG)
• p575 Rem12.4.26 "use a characterization" should be "uses a characterization"
• p585 last paragraph: "automaticatically" should be "automatically"
• p590 Thm12.4.36: "Use the" should be "The" (JG)
• p598 before Sec13.1: "In this ways" should be "In this way" (JG)
• p598 before Def13.1.1: "on all each scale" should be "on each scale" (JG)
• p601 after Exm13.1.6: "A realizer of size k" should be "In a realizer of size k" (JG), and "althoug" should be "although"
• p610 Exr13.1.5: delete final "}" (JG)
• p610 Exr13.1.6b: "assignment of interval" should be "assignment of intervals" (JG)
• p610 Exr13.1.14: "cut vertices" should be "cut-vertices" (JG)
• p612 Exr13.1.23d: missing final right parenthesis (JG)
• p626 Exr13.2.5: remove parentheses enclosing "Theorem 13.2.6" (JG)
• p639 Exr13.3.9: "lattices" should be "lattice" (JG)
• p650 Thm14.1.16: "a n-vertex graph" should be "an n-vertex graph" (JG)
• p672 Exr14.2.9a: and Exr14.2.10: add ending parenthesis (JG)
• p681 Thm14.3.15: "each possible copies" should be "each possible copy" (Michael Pelsmajer)
• p688 Exm14.3.24: "tested by expressed" should be "tested by expressing" (JG)
• p701 first full paragraph: "we final position" should be "the final position" (JG)
• p704 Def14.4.18: "has reach" should be "has reached" (JG)
• p711 after Def15.1.1: "offer" should be "often" (JG)
• p713 top: "approach in 18" should be "approach in Chapter 18" (JG)
• p715 after Exm15.1.11: "for for" should be "for" (JG)
• p715 Thm15.1.13: "the x's and y's" should be "x1,...,xr and y1,...,yr" (JG)
• p720 Lem15.1.24: "the the" should be "the" (JG)
• p725 Thm15.2.5: "obtaining by" should be "obtained by" (JG)
• p725 after Exm15.2.8: "using representing a digraph" should be "representing digraphs with multiple edges by" (JG)
• p730 Thm15.2.15: "have generate" should be "have generated" (JG)
• p733 Thm15.2.25: "it it achieves" should be "it achieves" (JG)
• p735 Thm15.2.27: "we known that" should be "we know that" (JG)
• p738 Exr15.2.6: "n-1byn-1 submatrix" should be "(n-1)-by-(n-1) submatrix" (JG)
• p752 Them15.3.31: "orthogonal to 1n" should be "orthogonal to 1n" (JG)
• p755 Exr15.3.5a: "recurrences relations" should be "recurrence relations" (JG)
• p762 before Lem15.4.10: "uses on" should be "makes use of" (JG)
• p765 Thm15.4.20 end: "then u'Q produces the received code word and "0 is position" should be "0 in position" (JG)
• p782 last line: "coordinate" should be "coordinates" (JG)
• p786 Thm16.1.8: "at clockwise" should be "and clockwise" (JG)
• p787 Thm16.1.10: "vertices claimed triple" should be "vertices in the claimed triple" (JG)
• p794 Thm16.1.28: delete the second copy of the references (JG)
• p199 Exr16.1.16: "prope drawings" should be "proper drawings" (JG)
• p805 Thm16.2.10: "distinct" should be "distinctness" (JG)
• p806 Thm16.2.15: "a the same" should be "the same" (JG)
• p826 third paragraph: "offering" should be "offer" (JG)
• p849 Thm17.2.18: "an Ks,t-free" should be "a Ks,t-free" (JG)
• p857 before Exercises: "extended it fo" should be "extended it for"
• p859 first paragraph: "digraph will" should be "digraph with" (JG)
• p859 second paragraph: "gave led" should be "led to" (JG)
• p859 third paragraph: "k - set" should be "k-set" (JG)
• p860 before bottom heading: "We Euler's" should be "We disprove Euler's" (JG)
• p861 Exm17.3.4: "interatively" should be "iteratively" (JG)
• p861 after Exm17.3.4: "and STS" should be "an STS" (JG)
• p866 top: "will developed" should be "will develop" (JG)
• p867 Exm17.3.18: "The resolvable" should be "Resolvability of" (JG)
• p868 before Def17.3.21: "efinition" should be "Definition" (JG)
• p869 Thm17.3.26: "each pairs" should be "each pair" (JG)
• p870 Thm17.3.27: "lables" should be "labels", and "value an reducing" should be "value and reducing" (JG)
• p895 Exm18.1.45: "interpret edge selected" should be "interpret edges selected" (JG)
• p896 Def18.1.46: "E-e" should be "E-e" (JG)
• p900 Exr18.1.40: "closure operating" should be "closure operation" (JG)
• p901 Exm18.2.2: "the ith of these partition matroid . . . ith partite set" should be "one of these partition matroids . . . the corresponding partite set" (JG)
• p914 Prop18.3.5: "nas another" should be "has another" (JG)
• p914 after Prop18.3.5: "the conversion" should be "the converse" (JG)
• p915 Def18.3.10: "that having" should be "that has" (JG)
• p921 Lem18.3.25: "is column matroid" should be "is the column matroid"
• p935 after Lem19.1.7: "This bring them" should be "This brings the variables for them" (JG)
• p938 last line: "procede" should be "proceed" (JG)
• p940 Thm19.1.14 last paragraph: "index of variable involved" should be "index of variables involved" (JG)
• p959 Exm19.2.11: "want maximize" should be "want to maximize" (JG)
• p981 Exr19.2.23: "a feasible solutions" should be "a feasible solution" (JG)
• p981 Exr19.2.24: "necessaryand" should be "necessary and" (JG)
• p983 before Def19.3.2: "if is vertices" should be "if its vertices (JG)
• p990 Prop19.3.19: "odd number of edge" should be "odd number of edges" (JG)
• p993 Exm19.3.25: "edges cuts" should be "edge cuts" (JG)
• p998 Thm19.3.36: "dual variable for" should be "dual variables form" (JG)
• p999 App19.3.38: "zero every except for the edges ts" should be "zero everwhere except on the edge ts, where it is 1" (JG)