On Tuza’s conjecture in dense graphs

Abstract

In 1982, Tuza conjectured that the size $\tau \left(G\right)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu \left(G\right)$ of a maximum set of edge-disjoint triangles of $G$. This conjecture was proved for several graph classes but it remains open even for split graphs. In this paper, we show Tuza’s conjecture for split graphs with minimum degree at least $\frac{3n}{5}$. We also show that $\tau \left(G\right)<\frac{28}{15}\nu \left(G\right)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$, and that $\tau \left(G\right)\le \frac{3}{2}\nu \left(G\right)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight.