A note on Ramsey Numbers for Books

A book of size N is the union of N triangles sharing a common edge. We show that the Ramsey number of a book of size N vs. a book of size M equals 2N+3 for all N>(10^6)M. Our proof is based on counting.Comment: 9 pages, submitted to Journal of Graph Theory in Aug 200

Dense H-free graphs are almost (Χ(H)-1)-partite

By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε>0 and sufficiently large n the following is true. Whenever G is an n-vertex graph with minimum degree δ(G)≥(1 − 3/3r−1 + ε)n, either G contains H, or we can delete f(n,H)≤Cn2−1/t edges from G to obtain an r-partite graph. Further, we are able to determine the correct order of magnitude of f(n,H) in terms of the Zarankiewicz extremal function

Odd Wheels in Graphs

AbstractFor k⩾1 the odd wheel of 2k+1 spokes, denoted by W2k+1, is the graph obtained from a cycle of length 2k+1 by adding a new vertex and joining it to all vertices of the cycle. In this paper it is shown that if a graph G of order n with minimum degree greater than 7n/12 is at least 4-chromatic then G contains an odd wheel with at most 5 spokes