Rainbow triangles and the Caccetta‐Häggkvist conjecture

Abstract

A famous conjecture of Caccetta and Häggkvist is that in a digraph on n vertices and minimum outdegree at least n/r there is a directed cycle of length r or less. We consider the following generalization: in an undirected graph on n vertices, any collection of n disjoint sets of edges, each of size at least n/r, has a rainbow cycle of length r or less. We focus on the case rgoodbreakinfix=3 and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. In our main result, whenever n is larger than a suitable polynomial in k, we determine the maximum number of edges in an n‐vertex edge‐colored graph where all color classes have size at most k and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.

Short rainbow cycles in graphs and matroids

“…. By (1) , the cycle matroid of G. We use the well-known fact that r N r N E G n ( ) + ( ) = | ( )| = 2 * . Therefore, since N has rank n N − 1, * has rank n + 1.…”

“…Given a graph and a coloring of , we say that a subgraph of is rainbow if no two edges of are of the same color. Aharoni (see [1]) recently proposed the following strengthening of the Caccetta‐Häggkvist conjecture.…”

“…By unmerging colors, C* is also a rainbow cocycle of G, so we are done. By(1), such an index j exists unless k E…”

Two remarks on the Caccetta-Häggkvist conjecture

“…Assuming negation, for every v ∈ A there exists w ∈ V (D) such that N + (w) = {v}. The w-term in the right hand side of (1) is then 1 2 , while the left hand side is at most 1 2 , and thus for the inequality to hold necessarily N − (v) = {w} and deg + (v) = 1. Namely, both in-degree and out-degree of v are 1.…”

“…Hompe also showed that this is false for digraphs with maximal out-degree larger than 2. In his construction, g(D) > ψ(D) ln 2+o (1) . In this paper we prove:…”