On Tuza's Conjecture in Dense Graphs

TL;DR

This work advances Tuza's conjecture in dense graphs by establishing a density-based boundary for split graphs ($\delta(G)\ge \frac{3}{5}n$) that ensures $\tau(G)\le 2\,\nu(G)$, and by deriving a new bound for dense tripartite graphs $\tau(G) \le \frac{n^2}{3(4m-n^2)}\,\nu(G)$, yielding $\tau(G)<1.8\,\nu(G)$ under a minimum-degree condition of $0.59n$. It also proves a tight, general bound for complete $4$-partite graphs, $\tau(G)\le \frac{3}{2}\,\nu(G)$, for graphs with at least five vertices. Collectively, these results extend the reach of Tuza-type bounds to several dense multipartite graph classes and introduce methods based on packing-extending via edge-colorings and $f$-factor techniques that may generalize further.

Abstract

In 1982, Tuza conjectured that the size $τ(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ν(G)$ 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 $τ(G) < \frac{28}{15}ν(G)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$. Finally, we show that $τ(G)\leq \frac{3}{2}ν(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight.

Figures (1)

• Figure 1: A complete 4-partite graph with $a = 4$, $b = 4$, $c = 4$, $d = 3$, and $x = 2$. Packing $P$ is formed by the solid edges, and packing $P'$ is formed by the dashed edges.

Theorems & Definitions (29)

• Conjecture 1: Tuza81
• Lemma 2
• proof
• Proposition 3: Vizing64, see also Diestel10
• Lemma 4
• proof
• Theorem 5
• Proposition 6: see also Bonamy2022
• proof
• Proposition 7: Feder2012