Properly colored and rainbow copies of graphs with few cherries

TL;DR

This paper proves Shearer's conjecture that graphs with few cherries must appear as properly colored or rainbow copies in certain edge colorings of complete graphs, establishing bounds and optimality.

Contribution

It confirms Shearer's conjecture for graphs with up to O(n^(4/3)) cherries and shows these bounds are tight, using advanced probabilistic methods.

Findings

Properly colored copies exist for graphs with up to O(n^(4/3)) cherries.

Rainbow copies can be found in bounded color frequency edge-colorings.

Bounds on the number of cherries are optimal up to a constant factor.

Abstract

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G with O(n^(4/3)) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G in every edge-coloring of K_n in which all colors appear bounded number of times. Our proofs combine a framework of Lu and Szekely for using the lopsided Lovasz local lemma in the space of random bijections together with some additional ideas.