In 1984, Fan gave a sufficient condition involving maximum degree of every pair of vertices at distance two for a graph to be Hamiltonian. Motivated by Fan's result, we say that an induced subgraph $H$ of a graph $G$ is $f$-heavy if for every pair of vertices $u,v\in V(H)$, $d_{ H}(u,v)=2$ implies that $\max\{ d(u),d(v)\}\geq n/2$. 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 $\mathcal{ R}$-$f$-\emph{ heavy} if $G$ is $R$-$f$-heavy for every $R\in \mathcal{ R}$. In this note we show that every 2-connected graph $G$ has a Hamilton cycle if $G$ is $\{ K_{ 1,3},P_7,D\}$-$f$-heavy or $\{ K_{ 1,3},P_7,H\}$-$f$-heavy, where $D$ is the deer and $H$ is the hourglass. Our result is a common generalization of previous theorems of Broersma et al. and Fan on Hamiltonicity of 2-connected graphs.
Categories: math.CO