A Note on Bipartite Subgraphs and Triangle-independent Sets

TL;DR

This paper improves an upper bound on the sum of the maximum size of a triangle-independent edge set and the minimum edge deletion set to make a graph bipartite, refining a longstanding conjecture.

Contribution

The authors provide a tighter bound of 4403n^2/15000, advancing the understanding of the relationship between triangle-independent sets and bipartite edge deletions.

Findings

Improved the upper bound from 5n^2/16 to 4403n^2/15000.

Confirmed the conjecture's bound can be significantly tightened.

Contributed to the theory of graph modifications related to triangles and bipartiteness.

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 $G$.