Tuza's 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, ${\nu }_3(G)$, 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 ${\tau }_3(G)>(1-o(1))\mathrm{\mid }G\mathrm{\mid }\mathrm{/}2$ and ${\nu }_3(G)<(1+\alpha )\mathrm{\mid }G\mathrm{\mid }\mathrm{/}4$.