Compatible powers of Hamilton cycles in dense graphs

The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph $G=\left(V,E\right)$, an incompatibility system $F$ over $G$ is a family $F={\left\{F_v\right\}}_{v\in V}$ such that for every $v\in V$, $F_v$ is a family of edge pairs $\left\{e,e^{\prime }\right\}\in \left(\frac{E\left(G\right)}{2}\right)$ with $e\cap e^{\prime }=\left\{v\right\}$. Moreover, for an integer $k\in N$, we say $F$ is $k$-bounded if for every vertex $v$ and its incident edge $e$, there are at most $k$ pairs in $F_v$ containing $e$. Krivelevich, Lee and Sudakov proved that there is an universal constant $\mu >0$ such that for every Dirac graph $G$ and every $\mu n$-bounded incompatibility system $F$ over $G$, there exists a Hamilton cycle $C\subseteq G$ where every pair of adjacent edges $e,e^{\prime }$ of $C$ satisfies $\left\{e,e^{\prime }\right\}\notin F_v$ for $\left\{v\right\}=e\cap e^{\prime }$. This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to $F$). We study high powers of Hamilton cycles in this context and show that for every $\gamma >0$ and $k\in N$, there exists a constant $\mu >0$ such that for sufficiently large $n\in N$ and every $\mu n$-bounded incompatibility system over an $n$-vertex graph $G$ with $\delta \left(G\right)\ge \left(\frac{k}{k+1}+\gamma \right)n$, there exists a compatible $k$th power of a Hamilton cycle in $G$. Moreover, we give a $\mu n$-bounded construction which has minimum degree $\frac{k}{k+1}n+\Omega \left(n\right)$ and contains no compatible $k$th power of a Hamilton cycle.