Combinatorics and Graph Theory
Springer Science & Business Media, Apr 3, 2009 - Mathematics - 381 pages
There are certain rules that one must abide by in order to create a successful sequel. — Randy Meeks, from the trailer to Scream 2 While we may not follow the precise rules that Mr. Meeks had in mind for s- cessful sequels, we have made a number of changes to the text in this second edition. In the new edition, we continue to introduce new topics with concrete - amples, we provide complete proofs of almost every result, and we preserve the book’sfriendlystyle andlivelypresentation,interspersingthetextwith occasional jokes and quotations. The rst two chapters, on graph theory and combinatorics, remain largely independent, and may be covered in either order. Chapter 3, on in nite combinatorics and graphs, may also be studied independently, although many readers will want to investigate trees, matchings, and Ramsey theory for nite sets before exploring these topics for in nite sets in the third chapter. Like the rst edition, this text is aimed at upper-division undergraduate students in mathematics, though others will nd much of interest as well. It assumes only familiarity with basic proof techniques, and some experience with matrices and in nite series. The second edition offersmany additionaltopics for use in the classroom or for independentstudy. Chapter 1 includesa new sectioncoveringdistance andrelated notions in graphs, following an expanded introductory section. This new section also introduces the adjacency matrix of a graph, and describes its connection to important features of the graph.
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2-coloring adjacency matrix adjacent algorithm assignment axiom of choice beads binomial coefficients bipartite graph blue color combinatorics complete graph compute contains contradiction convex countable cycle defined denote the number determine the number elements equivalent Erd˝os espousable Eulerian exactly example Exercise exists following theorem formula function Gale-Shapley algorithm graph G graph in Figure graph of order graph theory Hamiltonian identity implies induction infinite sets integer KĻonig’s Lemma labeled least Let G limit cardinal matrix nonempty nonnegative integer number of vertices objects obtain one-to-one ordinal pair partition partners path pattern inventory perfect matching permutation pigeonhole principle pigeons planar graph positive integer preference problem proof prove Ramsey numbers Ramsey theory real number sequence Show spanning tree stable marriage problem stable matching subgraph subset Suppose uncountable vertex of G weakly compact cardinal women