PACKING TRIANGLES IN WEIGHTED GRAPHS

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 <= 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 <= 2 nu* (where nu* is the fractional version of nu) and asked whether this is tight. We prove that tau <= 2 nu* - 1/root 6 root nu* and show that this bound is essentially best possible.