Compatible Powers of Hamilton Cycles in Dense Graphs

Motivated by the concept of transition system investigated by Kotzig in 1968, Krivelevich, Lee and Sudakov proposed a more general notion of incompatibility system to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph $G=(V,E)$, an {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V}$ such that for every $v\in V$, $F_v$ is a family of edge pairs in $\{\{e,e'\}: e\ne e'\in E, e\cap e'=\{v\}\}$. An incompatibility system $\mathcal{F}$ is \emph{$\Delta$-bounded} if for every vertex $v$ and every edge $e$ incident with $v$, there are at most $\Delta$ pairs in $F_v$ containing $e$. A subgraph $H$ of $G$ is \emph{compatible} (with respect to $\mathcal{F}$) if every pair of adjacent edges $e,e'$ of $H$ satisfies $\{e,e'\} \notin F_v$, where $v=e\cap e'$. Krivelevich, Lee and Sudakov proved that there is an universal constant $\mu>0$ such that for every $\mu n$-bounded incompatibility system $\mathcal{F}$ over a Dirac graph, there exists a compatible Hamilton cycle, which resolves a conjecture of H\"{a}ggkvist from 1988. We study high powers of Hamilton cycles in this context and show that for every $\gamma>0$ and $k\in\mathbb{N}$, there exists a constant $\mu>0$ such that for sufficiently large $n\in\mathbb{N}$ and every $\mu n$-bounded incompatibility system over an $n$-vertex graph $G$ with $\delta(G)\ge(\frac{k}{k+1}+\gamma)n$, there exists a compatible $k$-th power of a Hamilton cycle in $G$. Moreover, we give a construction which has minimum degree $\frac{k}{k+1}n+\Omega(n)$ and contains no compatible $k$-th power of a Hamilton cycle.