Combinatorics and Graph Theory
Springer Science & Business Media, Jul 19, 2000 - Mathematics - 228 pages
Three things should be considered: problems, theorems, and applications. - Gottfried Wilhelm Leibniz, Dissertatio de Arte Combinatoria, 1666 This book grew out of several courses in combinatorics and graph theory given at Appalachian State University and UCLA in recent years. A one-semester course for juniors at Appalachian State University focusing on graph theory covered most of Chapter 1 and the first part of Chapter 2. A one-quarter course at UCLA on combinatorics for undergraduates concentrated on the topics in Chapter 2 and included some parts of Chapter I. Another semester course at Appalachian State for advanced undergraduates and beginning graduate students covered most of the topics from all three chapters. There are rather few prerequisites for this text. We assume some familiarity with basic proof techniques, like induction. A few topics in Chapter 1 assume some prior exposure to elementary linear algebra. Chapter 2 assumes some familiarity with sequences and series, especially Maclaurin series, at the level typically covered in a first-year calculus course. The text requires no prior experience with more advanced subjects, such as group theory.
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2-coloring algorithm apply axiom of choice beads binomial coefficients bipartite graph blue chromatic polynomial combinatorics complete graph components compute connected contradiction cut vertex cycle defined denote edges of G elements encoding exactly example Exercise exists following theorem formula function assignment graph G graph in Figure graph of order graph theory identity implies induction infinite sets knights Konig's Lemma large cardinals least Let G limit cardinal Marv Mathematics matrix Milt monochromatic natural numbers nonempty nonplanar number of edges number of vertices objects one-to-one ordinal pair partition path perfect matching permutation pigeonhole principle pigeons planar graph planar representation problem proof properties prove Ramsey numbers Ramsey theory Ramsey's Theorem real number regular result sequence Show spanning tree stable matching Stirling numbers subset Suppose tree of order vertex Walda Wanda weakly compact cardinal well-ordered Wilma Winny
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