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(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 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.

Rainbow matchings in Dirac bipartite graphs

“…The existence of rainbow copies of H in k ‐bounded edge colorings of K n was studied in , provided that k = O ( n /Δ 2 ). In , it was observed that similar techniques allow to replace K n by a graph G with , for a sufficiently small constant c > 0.…”

“…Sudakov and Volec showed that there exist a graph H with maximum degree at most Δ and a 3.9 n /Δ 2 ‐bounded edge coloring of K n which does not contain a rainbow copy of H . Therefore this theorem is also tight up to constant factors.…”

Long properly coloured cycles in edge‐coloured graphs

“…A hypergraph generalisation of finding properly coloured Hamilton cycle in locally ‐bounded edge‐coloured complete graphs has also been studied by Dudek, Frieze and Ruciński as well as Dudek and Ferrara . Recently, Sudakov and Volec proved that every locally ‐bounded edge‐coloured contains all properly coloured graphs with at most paths of length two. This proved a conjecture of Shearer as well as improves results of Alon, Jiang, Miller, Pritikin and Böttcher, Kohayakawa and Procacci .…”