Extremal Graphs without Large Forbidden Subgraphs

Abstract

The theory of extremal graphs without a fixed set of forbidden subgraphs is well developed. However, rather little is known about extremal graphs without forbidden subgraphs whose orders tend to ∞ with the order of the graph. In this note we deal with three problems of this latter type. Let L be a fixed bipartite graph and let L + E m be the join of L with the empty graph of order m. As our first problem we investigate the maximum of the size e(G n ) of a graph G n (i.e. a graph of order n) provided G n ⊅L + E[ cn , where c > 0 is a constant. In our second problem we study the maximum of e(G n ) if G n ⊅K 2 (r,cn) and G n ⊅ K 3 . The third problem is of a slightly different nature. Let C k (t) be obtained from a cycle C k by multiplying each vertex by t. We shall prove that if c > 0 then there exists a constant l(c) such that if G n ⊅C k (t) for k = 3, 5, 2l(c) + 1, then one can omit [cn 2 ] edges from G n so that the obtained graph is bipartite, provided n > n 0 (c, t).