Hamilton Cycles in Claw-Heavy Graphs with Fan-Type Condition Restricted to Two Induced Subgraphs

Abstract

A graph $G$ on $n\ge 3$ vertices is called claw-heavy if every induced claw of $G$ has a pair of nonadjacent vertices such that their degree sum is at least $n$. We say that a subgraph $H$ of $G$ is $f$-heavy if $max\{d(x),d(y)\}\ge \frac{n}{2}$ for every pair of vertices $x,y\in V(H)$ at distance $2$ in $H$. For a given graph $R$, $G$ is called $R$-$f$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f$-heavy. For a family $\mathcal{R}$ of graphs, $G$ is called $\mathcal{R}$-$f$-heavy if $G$ is $R$-$f$-heavy for every $R\in \mathcal{R}$. In this paper, we show that every $2$-connected claw-heavy graph is hamiltonian if $G$ is $\{P_7,D\}$-$f$-heavy, or $\{P_7,H\}$-$f$-heavy, where $D$ is a deer and $H$ is a hourglass. Our result is a common generalization of previous theorems of Broersma et al. and Fan on hamiltonicity of $2$-connected graphs.