Aharoni's rainbow cycle conjecture holds up to an additive constant

Abstract

 In 2017, Ron Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most the ceiling of n/r.  

I will begin with a summary of recent progress on Aharoni's conjecture based on a new survey article of Katie Clinch, Jackson Goerner, Freddie Illingworth, and myself.  I will then sketch a proof that Aharoni's conjecture holds up to an additive constant for each fixed r.  The last result is joint work with Patrick Hompe.