  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.
