A Note on Bipartite Subgraphs and Triangle-independent Sets

Let $\alpha_{1} (G)$ denote the maximum size of an edge set that contains at most one edge from each triangle of $G$. Let $\tau_{B} (G)$ denote the minimum size of an edge set whose deletion makes $G$ bipartite. It was conjectured by Lehel and independently by Puleo that $\alpha_{1} (G) + \tau_{B} (G) \le n^2/4$ for every $n$-vertex graph $G$. Puleo showed that $\alpha_{1} (G) + \tau_{B} (G) \le 5n^2/16$ for every $n$-vertex graph $G$. In this note, we improve the bound by showing that $\alpha_{1} (G) + \tau_{B} (G) \le 4403n^2/15000$ for every $n$-vertex graph $G$.