Packing triangles in weighted graphs

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 $\tau \le 2\nu$. 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 $\tau \le 2{\nu }^{\ast }$ (where ${\nu }^{\ast }$ is the fractional version of $\nu$), and asked if this is tight. We prove that $\tau \le 2{\nu }^{\ast }-\frac{1}{\sqrt{6}}\sqrt{{\nu }^{\ast }}$ and show that this bound is essentially best possible.

v2: 20 pages, corrected version (from 2013) following referee reports