Compatible Powers of Hamilton Cycles in Dense Graphs

TL;DR

This paper extends the study of Hamiltonicity robustness in dense graphs with incompatibility constraints, proving the existence of compatible high powers of Hamilton cycles under certain degree conditions.

Contribution

It generalizes previous results by establishing conditions for the existence of compatible k-th powers of Hamilton cycles in graphs with incompatibility systems.

Findings

Existence of compatible k-th power Hamilton cycles under degree conditions

Construction of graphs with high minimum degree lacking such cycles

Extension of robustness results to higher powers of Hamilton cycles

Abstract

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.