Graph tilings in incompatibility systems

TL;DR

This paper investigates conditions under which large graphs with certain incompatibility constraints contain perfect subgraphs isomorphic to a given graph, extending previous results with a new lattice-based absorption proof technique.

Contribution

It establishes a new threshold for the existence of compatible H-factors in graphs with incompatibility systems, using a novel lattice-based absorption method.

Findings

Proves existence of compatible H-factors under minimum degree conditions.

Identifies a dichotomy based on the chromatic number of H.

Provides counterexamples showing sharpness of the results.

Abstract

An \emph{incompatibility system} $(G,\mathcal{F})$ consists of a graph $G$ and a family $\mathcal{F}=\{F_v\}_{v\in V(G)}$ over $G$ with $F_v\subseteq \{\{e,e'\}\in {E(G)\choose 2}: e\cap e'=\{v\}\}$. We say that two edges $e,e'\in E(G)$ are \emph{incompatible} if $\{e,e'\}\in F_v$ for some $v\in V(G)$, and otherwise \emph{compatible}. A subgraph $H$ of $G$ is \emph{compatible} if every pair of edges in $H$ are compatible. An incompatibility system $(G,\mathcal{F})$ is \emph{$\Delta$-bounded} if for any vertex $v$ and any edge $e$ incident with $v$, there are at most $\Delta$ members of $F_v$ containing $e$. This notion was partly motivated by a concept of transition system introduced by Kotzig in 1968, and first formulated by Krivelevich, Lee and Sudakov to study the robustness of Hamiltonicity of Dirac graphs. We prove that for any $\alpha>0$ and any graph $H$ with $h$ vertices, there exists a constant $\mu>0$ such that for any sufficiently large $n$ with $n\in h\mathbb{N}$, if $G$ is an $n$-vertex graph with $\delta(G)\ge(1-\frac{1}{\chi^*(H)}+\alpha)n$ and $(G,\mathcal{F})$ is a $\mu n$-bounded incompatibility system, then there exists a compatible $H$-factor in $G$, where the value $\chi^*(H)$ is either the chromatic number $\chi(H)$ or the critical chromatic number $\chi_{cr}(H)$ and we provide a dichotomy as in the K\"{u}hn--Osthus result. Moreover, we give examples $H$ for which there exists an $\mu n$-bounded incompatibility system $(G, \mathcal{F})$ with $n\in h\mathbb{N}$ and $\delta(G)\ge(1-\frac{1}{\chi^*(H)}+\frac{\mu}{2})n$ such that $G$ contains no compatible $H$-factor. Unlike in the previous work of K\"{u}hn and Osthus on embedding $H$-factors, our proof uses the lattice-based absorption method.