Rainbow Hamilton cycles and lopsidependency

Abstract

The Lovász Local Lemma is a powerful probabilistic tool used to prove the existence of combinatorial structures which avoid a set of constraints. A standard way to apply the local lemma is to prove that the set of constraints satisfy a lopsidependency condition and obtain a lopsidependency graph. For instance, Erdős and Spencer used this framework to posit the existence of Latin transversals in matrices provided no symbol appears too often in the matrix.The local lemma has been used in various ways to infer the existence of rainbow Hamilton cycles in complete graphs when each colour is used at most O(n) times. However, the existence of a lopsidependency graph for Hamilton cycles has neither been proved nor refuted. All previous approaches have had to prove a variant of the local lemma or reduce the problem of finding Hamilton cycles to finding another combinatorial structure, such as Latin transversals. In this paper, we revisit the question of whether or not Hamilton cycles have a lopsidependency graph and give a positive answer for this question. We also use the resampling oracle framework of Harvey and Vondrák to give a polynomial time algorithm for finding rainbow Hamilton cycles in complete graphs.