On Tuza's Conjecture in Dense Graphs

TL;DR

This paper advances understanding of Tuza's Conjecture in dense graphs by proving it for specific classes and establishing tight bounds for complete 4-partite graphs using probabilistic and combinatorial methods.

Contribution

The paper extends Tuza's Conjecture validity to split and tripartite graphs and provides a tight bound for complete 4-partite graphs, using probabilistic and combinatorial techniques.

Findings

Tuza's Conjecture holds for graphs with minimum degree at least 7n/8.

The conjecture is valid for split graphs with minimum degree at least 3n/5.

For complete 4-partite graphs, τ(G) ≤ 3/2 ν(G), and this bound is tight.

Abstract

In 1982, Tuza conjectured that the size $\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. This conjecture was proved for several graph classes. In this paper, we present three results regarding Tuza's Conjecture for dense graphs. By using a probabilistic argument, Tuza proved its conjecture for graphs on $n$ vertices with minimum degree at least $\frac{7n}{8}$. We extend this technique to show that Tuza's conjecture is valid for split graphs with minimum degree at least $\frac{3n}{5}$; and that $\tau(G) < \frac{28}{15}\nu(G)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$. Finally, we show that $\tau(G)\leq \frac{3}{2}\nu(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight.