Tuzas Conjecture is Asymptotically Tight for Dense Graphs

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, $ u_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 $ u_3(G)<(1+alpha)|G|/4$.