tailieunhanh - Báo cáo toán học: "Graphs with many copies of a given subgraph"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Graphs with many copies of a given subgraph. | Graphs with many copies of a given subgraph Vladimir Nikiforov Department of Mathematical Sciences University of Memphis Memphis TN 38152 vnikifr@ Submitted Oct 8 2007 Accepted Mar 3 2008 Published Mar 12 2008 Mathematics Subject Classification 05C35 Abstract Let c 0 and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a blow-up of H with r 1 vertex classes of size _cr ln nJ and one vertex class of size greater than n1-cr . A similar result holds for induced copies of H . Main results Suppose that a graph G of order n contains cnr copies of a given subgraph H on r vertices. How large blow-up of H must G contain When H is an r-clique this question was answered in 3 G contains a complete r-partite graph with r 1 parts of size _cr In nJ and one part larger than n1-cr 1. The aim of this note is to answer this question for any subgraph H. First we define precisely a blow-up of a graph given a graph H of order r and positive integers x1 . xr we write H x15. xr for the graph obtained by replacing each vertex u 2 V H with a set Vu of size xu and each edge uv 2 E H with a complete bipartite graph with vertex classes Vu and Vv. Theorem 1 Let 2 r n ln n _1 r2 c 1 4 H be a graph of order r and G be a graph of order n. If G contains more than cnr copies of H then G contains a copy of H s . s t where s _c ln nJ and t nx cC . To state a similar theorem for induced subgraphs we need a proper modification of H x1 . xr we say that a graph X is of type H x1 . xr if X is obtained from H x1 . xr by adding some possibly zero edges within the sets Vu u 2 V H . Theorem 2 Let 2 r n ln n _1 r2 c 1 4 H be a graph of order r and G be a graph of order n. If G contains more than cnr induced copies of H then G contains an induced subgraph of type H s . s t where s _cr ln nJ and t n c . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N6 1 Remarks - The relations between c and n in Theorems 1 and 2 need some explanation. First for .