Packing triangles in weighted graphs
Guillaume Chapuy, Matt DeVos, Jessica McDonald, Bojan Mohar, Diego, Scheide

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 , the maximum size of a set of edge-disjoint triangles and minimum size of a set of edges meeting all triangles satisfy . 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 (where is the fractional version of ), and asked if this is tight. We prove that and show that this bound is essentially best possible.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · graph theory and CDMA systems
