Triangle-independent sets vs. cuts

Abstract

A set of edges $T$ in a graph $G$ is triangle-independent if $T$ contains at most one edge from each triangle in $G$. Let ${\alpha }_1(G)$ denote the maximum size of the triangle-independent set in $G$, and let ${\tau }_B(G)$ denote minimum size of a set $F\subseteq E(G)$ such that $G\setminus F$ is bipartite. We prove that $${\alpha }_1(G)+{\tau }_B(G)\le \frac{|V(G){|}^2}{4},$$ verifying a conjecture due to Lehel, and independently Puleo, and a slightly weaker conjecture of Erdős, Gallai and Tuza. Further, we characterize the graphs which attain the equality.