Properly colored and rainbow copies of graphs with few cherries

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 Kn 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(n4/3) 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 Kn in which all colors appear bounded number of times.Our proofs combine a framework of Lu and Székely for using the lopsided Lovász local lemma in the space of random bijections together with some additional ideas.