On 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⌉. 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 Ω(klogk).