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 ${\alpha }_1(G)+{\tau }_B(G)\le n^2\mathrm{/}4$ for every $n$-vertex graph $G$. Puleo showed that $\alpha_{1} (G) + \tau_{B} (G) \le 5n^2/16$forevery$n$-vertexgraph$G$. In this note, we improve the bound by showing that ${\alpha }_1(G)+{\tau }_B(G)\le 4403n^2\mathrm{/}15000$ for every $n$-vertex graph $G$.