Graphs with $\alpha _1$ and $\tau _1$ Both Large

Abstract

Given a graph G, 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 an edge set containing at most one edge from each triangle of G. Erdős, Gallai, and Tuza introduced several problems with the unifying theme that $\alpha _1(G)$ and $\tau _1(G)$ cannot both be “very large”; the most well-known such problem is their conjecture that $\alpha _1(G) + \tau _1(G) \le \left|{V(G)}\right|^2/4$, which was proved by Norin and Sun. We consider three other problems within this theme (two introduced by Erdős, Gallai, and Tuza, another by Norin and Sun), all of which request an upper bound either on $\min \{\alpha _1(G), \tau _1(G)\}$ or on $\alpha _1(G) + k\tau _1(G)$ for some constant k, and prove the existence of graphs for which these quantities are “large”.