Cubic graph

Abstract: The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer\'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader.

Abstract: Lecture 1 The basic statement of extremal graph theory is Mantel's theorem, proved in 1907, which states that any graph on n vertices with no triangle contains at most n 2 /4 edges. This is clearly best possible, as one may partition the set of n vertices into two sets of size n/2 and n/2 and form the complete bipartite graph between them. This graph has no triangles and n 2 /4 edges. As a warm-up, we will give a number of different proofs of this simple and fundamental theorem. Theorem 1 (Mantel's theorem) If a graph G on n vertices contains no triangle then it contains at most n 2 4 edges. First proof Suppose that G has m edges. Let x and y be two vertices in G which are joined by an edge. If d(v) is the degree of a vertex v, we see that d(x) + d(y) ≤ n. This is because every vertex in the graph G is connected to at most one of x and y. Note now that x d 2 (x) = xy∈E (d(x) + d(y)) ≤ mn. On the other hand, since x d(x) = 2m, the Cauchy-Schwarz inequality implies that x d 2 (x) ≥ (x d(x)) 2 n ≥ 4m 2 n. Therefore 4m 2 n ≤ mn, and the result follows. 2 Second proof We proceed by induction on n. For n = 1 and n = 2, the result is trivial, so assume that we know it to be true for n − 1 and let G be a graph on n vertices. Let x and y be two adjacent vertices in G. As above, we know that d(x)+d(y) ≤ n. The complement H of x and y has n−2 vertices and since it contains no triangles must, by induction, have at most (n − 2) 2 /4 edges. Therefore, the total number of edges in G is at most e(H) + d(x) + d(y) − 1 ≤ (n − 2) 2 4 + n − 1 = n 2 4 , where the −1 comes from the fact that we count the edge between x and y twice. 2

Abstract: It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} . The method is then generalized to obtain upper bounds on the capacity of an arbitrary graph. A well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases. Several results are obtained on the capacity of special graphs; for example, the Petersen graph has capacity four and a self-complementary graph with n points and with a vertex-transitive automorphism group has capacity \sqrt{5} .