On a Conjecture of Erd\H{o}s, Gallai, and Tuza

Erd\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex 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 a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\alpha_1(G) + \tau_1(G) \leq n^2/4$? We have two main results. We first obtain the upper bound $\alpha_1(G) + \tau_1(G) \leq 5n^2/16$, as a partial result towards the Erd\H{o}s--Gallai--Tuza conjecture. We also show that always $\alpha_1(G) \leq n^2/2 - m$, where $m$ is the number of edges in $G$; this bound is sharp in several notable cases.