Rainbow triangles and the Caccetta‐Häggkvist conjecture

Abstract

AbstractA famous conjecture of Caccetta and Häggkvist is that in a digraph on vertices and minimum outdegree at least n/r there is a directed cycle of length or less. We consider the following generalization: in an undirected graph on vertices, any collection of disjoint sets of edges, each of size at least n/r, has a rainbow cycle of length or less. We focus on the case and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. In our main result, whenever is larger than a suitable polynomial in , we determine the maximum number of edges in an ‐vertex edge‐colored graph where all color classes have size at most and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.