Tuza's Conjecture is Asymptotically Tight for Dense Graphs

TL;DR

This paper demonstrates that Tuza's conjecture, which relates triangle covering and packing in graphs, is asymptotically tight for dense graphs, providing constructions that nearly reach the conjectured bounds.

Contribution

It proves that for dense graphs, the ratio of triangle cover to packing can approach the conjectured limit, disproving a previous conjecture and showing the bounds are asymptotically tight.

Findings

Existence of arbitrarily large dense graphs with high triangle cover ratios.

Construction of graphs where triangle packing is significantly smaller than the cover.

Disproof of Yuster's conjecture on the bounds of triangle cover and packing ratios.

Abstract

An old conjecture of Zs. Tuza says that for any graph $G$, the ratio of the minimum size, $\tau_3(G)$, of a set of edges meeting all triangles to the maximum size, $\nu_3(G)$, of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive $\alpha$ there are arbitrarily large graphs $G$ of positive density satisfying $\tau_3(G)>(1-o(1))|G|/2$ and $\nu_3(G)<(1+\alpha)|G|/4$.