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

Abstract

A graph \(G\) on \(n \geq 3\) 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 \(\max\{d(x), d(y)\} \geq \frac{n}{2}\) 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.