Compatible Hamilton cycles in random graphs

Abstract

AbstractA 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 , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an incompatibility system over G is a family where for every , the set Fv is a set of unordered pairs . 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 if every pair of incident edges of C satisfies . 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 such that the random graph with is asymptotically almost surely such that for any μnp‐bounded incompatibility system over G, there is a Hamilton cycle in G compatible with . We also prove that for larger edge probabilities , the parameter μ can be taken to be any constant smaller than . These results imply in particular that typically in G(n, p) for , 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