On the maximal number of certain subgraphs inK r -free graphs

Abstract

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) ≤ H(T 2(n)), whereT 2(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n ? max(m, r ? 1), then there exists a complete (r ? 1)-partite graphG* withn vertices such thatK(G) ≤ K(G*) holds for everyK r -free graphG withn vertices. In particular, in the class of allK r -free graphs withn vertices the complete balanced (r ? 1)-partite graphT r?1(n) has the largest number of subgraphs isomorphic toK t (t < r),C 4,K 2,3. These generalize some theorems of Turan, Erdos and Sauer.