Compatible Hamilton cycles in random graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability p logn n , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph G = (V,E), an incompatibility system F over G is a family F = {Fv}v∈V where for every v ∈ V , the set Fv is a set of unordered pairs Fv ⊆ {{e, e′} : e 6= e′ ∈ E, e ∩ e′ = {v}}. An incompatibility system is ∆-bounded if for every vertex v and an edge e incident to v, there are at most ∆ pairs in Fv containing e. We say that a cycle C in G is compatible with F if every pair of incident edges e, e′ of C satisfies {e, e′} / ∈ Fv. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant μ > 0 such that the random graph G = G(n, p) with p(n) logn n is asymptotically almost surely such that for any μnp-bounded incompatibility system F over G, there is a Hamilton cycle in G compatible with F . We also prove that for larger edge probabilities p(n) log 8 n n , the parameter μ can be taken to be any constant smaller than 1 − 1 √ 2 . These results imply in particular that typically in G(n, p) for p logn n , for any edge-coloring in which each color appears at most μnp times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge-coloring of such a random graph in which each color appears on at most μnp edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle).