How to make a graph bipartite

Let G be a graph with vertex set V(G) and edge set E(G), respectively. The set of vertices adjacent to an x e V(G) is denoted by F(x), and the degree of x is d(x) = I F(x)l . For any subset V 'g V(G), let G[ V] denote the subgraph of G induced by the vertices of V' . Further, let K n stand for the complete graph on n vertices. It is easily seen (e.g ., Erdös [7]) that every graph G with n vertices and in edges contains a bipartite subgraph H such that iE(H)l >IE(G)1/2 = m/2, i .e., every graph can be made bipartite by the omission of at most half of its edges . Erdös and Lovász proved that if G has no triangle, then it can be made bipartite by the omission of m/2 _ M 2/3 (log m)` edges . On the other hand, Erdös [9] showed by the probability method that for every r, there is a graph G with no cycle of length less than r which cannot be made bipartite by the omission of fewer than in edges. The best exponent in m' is not known even for r=3, but s . approaches 0 as r becomes large . However, the graphs constructed in [9] are "sparse" (i.e ., in = 0(n')), and the aim of this paper is to show that much stronger results can be obtained if we assume that our graph G is not sparse. We will restrict our attention to families of graphs not containing some so-called,lbrbidden subgraph E (Such graphs are also said to be F-free .) In particular, for triangle-free graphs, i .e ., when F= K3 , we will prove the following . 86 0095-8956,,88 53 .00