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≫lognn, 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≠e′∈E,e∩e′={v}}. An incompatibility system is Δ-bounded if for every vertex v and an edge e incident to v, there are at most Δ 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)≫lognn 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)≫log8nn, the parameter μ can be taken to be any constant smaller than 1−12. These results imply in particular that typically in G(n, p) for p≫lognn, 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). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 533–557, 2016