Compatible Hamilton cycles in graphs with large minimum degree

Abstract

The renowned theorem of Dirac states that if $G$ is a graph with minimum degree at least $n/2$ then $G$ has a Hamilton cycle. A natural generalisation asks what properties of an edge-colouring of $G$ guarantee the existence of a properly edge-coloured Hamilton cycle in $G$. This concept can be further generalised as follows: an \emph{incompatibility system} for $G$ is a set~$\mathcal{F}$ of `forbidden' pairs of adjacent edges, that is, $\mathcal{F}\subseteq \{\{uv,vw\}\in \binom{E(G)}2\}$. A cycle in $G$ is then \emph{compatible} if no two of its edges form a pair in $\mathcal{F}$. The system $\mathcal{F}$ is called \emph{$μn$-bounded} if for all $v\in V(G)$ and $uv\in E(G)$, there are at most $μn$ pairs $\{uv,vw\}\in \mathcal{F}$. How small must $μ$ be to guarantee the existence of a compatible Hamilton cycle in $G$? Krivelevich, Lee and Sudakov showed that $μ=10^{-16}$ suffices (for $n$ large), while an example of Bollobás and Erdős shows that $μ\leq 1/4$ is necessary. We significantly reduce this gap for large graphs of minimum degree at least $(1/2+\varepsilon)n$, by showing that $μ=1/8$ suffices but $μ\leq 1/6$ is necessary for such graphs. In fact, we give more precise bounds which are functions of $δ(G)/n$.

Figures (2)

• Figure 1: The bounds on $\mu\coloneqq \mu(\delta)$. The feasible region is shaded.
• Figure 2: An absorbing path for $(x_1x_1',x_2x_2')$.

Theorems & Definitions (47)

• Theorem 1.1: Krivelevich, Lee and Sudakov KLS
• Definition 1.3
• Proposition 1.4
• Proposition 1.5
• Theorem 1.6
• Theorem 1.7: Almost regular graphs
• Corollary 1.8
• Corollary 1.9
• proof : Proof of \ref{['prop:upper_bound_reg']}
• proof : Proof of \ref{['prop:upper_bound_nonreg']}