Packing triangles in weighted graphs

TL;DR

This paper extends Tuza's conjecture to weighted multigraphs, proving bounds on triangle packing and covering that improve previous results and are nearly optimal.

Contribution

It generalizes Tuza's conjecture to weighted multigraphs and establishes nearly tight bounds on triangle packing and covering.

Findings

Proves $ au \,\leq 2\nu^* - \frac{1}{\sqrt{6}}\sqrt{\nu^*}$ for weighted multigraphs.

Shows the bound is essentially tight.

Answers Krivelevich's question on the tightness of the bound.

Abstract

Tuza conjectured that for every graph $G$, the maximum size $\nu$ of a set of edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich who proved that $\tau \leq 2\nu^*$ (where $\nu^*$ is the fractional version of $\nu$), and asked if this is tight. We prove that $\tau \leq 2\nu^*-\frac{1}{\sqrt{6}}\sqrt{\nu^*}$ and show that this bound is essentially best possible.