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(n) » log n/n, the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph (Formula presented.), an incompatibility system F over G is a family F = {F_v}_v∈V where for every _v∈V, the set F_v is a set of unordered pairs p(n) » log n/n. 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 F_v 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'}. 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 F 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 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 » log n/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).