Tuza's Conjecture is Asymptotically Tight for Dense Graphs

Abstract

An old conjecture of Z. Tuza says that for any graphG, the ratio of the minimum size, τ3(G), of a set of edges meeting all triangles to the maximum size, ν3(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 α there are arbitrarily large graphsGof positive density satisfying τ3(G) > (1 −o(1))|G|/2 and ν3(G) < (1 + α)|G|/4.