Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

Abstract

For a digraph G and v∈V(G), let δ+(v) be the number of out-neighbors of v in G. The Caccetta-Häggkvist conjecture states that for all k≥1, if G is a digraph with n=|V(G)| such that δ+(v)≥k for all v∈V(G), then G contains a directed cycle of length at most ⌈n/k⌉. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size at least k, has a rainbow cycle of length at most ⌈n/k⌉. Let us call (α,β)triangular if every simple edge-colored graph on n vertices with at least αn color classes, each with at least βn edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following:•(9/8,1/3) is triangular;•(1,2/5) is triangular. In this paper, we improve those bounds, showing the following:•(1.1077,1/3) is triangular;•(1,0.3988) is triangular. Our methods give results for infinitely many pairs (α,β), including β<1/3; we show that (1.3481,1/4) is triangular.