A Note on Bipartite Subgraphs and Triangle-independent Sets

Abstract

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 $G$.

An error in the first version of the paper was corrected

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