Abstract: Erdős, Gallai, and Tuza posed the following problem: given an n-vertex graph $G$, let $τ_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $α_1(G)$ 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 $α_1(G) + τ_1(G) ≤ n^2/4$? A positive answer would generalize Mantel's Theorem, which states that the largest possible number of edges in a triangle-free graph is $n^2/4$. In this talk, we show three main results. We first obtain the upper bound $α_1(G) + τ_1(G) ≤ 5n^2/16$, as a partial result towards the Erdős--Gallai--Tuza conjecture. We then study the properties of a minimal counterexample to the conjecture, showing that any minimal counterexample has "dense edge cuts" and in particular has minimum degree greater than $n/2$. This reconciles the two different formulations of the conjecture found in the literature, since it implies that the Erdős--Gallai--Tuza conjecture holds for all graphs if and only if it holds for graphs for which every edge lies in a triangle. Finally, we show that the conjecture holds for all graphs which contain no induced subgraph isomorphic to $K_4^-$, the graph obtained from $K_4$ by removing an edge.