Compatible Hamilton cycles in random graphs

Abstract: 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 $p \gg \frac{\log n}{n}$, the random graph $G(n,p)$ is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph $G=(V,E)$, an {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}={F_v}_{v\in V}$ where for every $v\in V$, the set $F_v$ is a set of unordered pairs $F_v \subseteq {{e,e'}: e\ne e'\in E, e\cap e'={v}}$. An incompatibility system is {\em $\Delta$-bounded} if for every vertex $v$ and an edge $e$ incident to $v$, there are at most $\Delta$ pairs in $F_v$ containing $e$. We say that a cycle $C$ in $G$ is {\em compatible} with $\mathcal{F}$ if every pair of incident edges $e,e'$ of $C$ satisfies ${e,e'} \notin F_v$. 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 $\mu>0$ such that the random graph $G=G(n,p)$ with $p(n) \gg \frac{\log n}{n}$ is asymptotically almost surely such that for any $\mu np$-bounded incompatibility system $\mathcal{F}$ over $G$, there is a Hamilton cycle in $G$ compatible with $\mathcal{F}$. We also prove that for larger edge probabilities $p(n)\gg \frac{\log^8n}{n}$, the parameter $\mu$ can be taken to be any constant smaller than $1-\frac{1}{\sqrt 2}$. These results imply in particular that typically in $G(n,p)$ for $p \gg \frac{\log n}{n}$, for any edge-coloring in which each color appears at most $\mu np$ times at each vertex, there exists a properly colored Hamilton cycle.