Extremal aspects of the Erdos-Gallai-Tuza conjecture

Erdos, Gallai, and Tuza posed the following problem: given an n-vertex graph C. let tau(1) (G) denote the smallest size of a set of edges whose deletion makes G triangle-free, and let alpha(1) (G) denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that alpha(1) (G) + tau(1) (G) <= n(2)/4? We also consider a variant on this conjecture: if tau(B)(G) is the smallest size of an edge set whose deletion makes G bipartite, does the stronger inequality alpha(1) (G) + tau(B) (G) <= n(2)/4 always hold? By considering the structure of a minimal counterexample to each version of the conjecture, we obtain two main results. Our first result states that any minimum counterexample to the original ErdO's-Gallai-Tuza Conjecture has "dense edge cuts", and in particular has minimum degree greater than n/2. This implies that the conjecture holds for all graphs if and only if it holds for all triangular graphs (graphs where every edge lies in a triangle). Our second result states that alpha(1) (G) + tau(B)(G) <= n(2)/4 whenever G has no induced subgraph isomorphic to K-4(-), the graph obtained from the complete graph K-4 by deleting an edge. Thus, the original conjecture also holds for such graphs. (C) 2015 Elsevier B.V. All rights reserved.