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A Note on Bipartite Subgraphs and Triangle-independent Sets

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