On Aharoni’s rainbow generalization of the Caccetta–Häggkvist conjecture

Abstract

For a digraph G and v is an element of V(G), let delta(+)(v) be the number of out-neighbors of v in G. The Caccetta-Haggkvist conjecture states that for all k >= 1, if G is a digraph with n = |V(G)| such that delta(+)(v) >= k for all v is an element of V(G), then G contains a directed cycle of length at most [n/k]. In Aharoni et al. (2019), Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size k, has a rainbow cycle of length at most.n/k.. In this paper, we prove this conjecture if each color class has size Omega(k log k).