Fan-type degree condition restricted to triples of induced subgraphs ensuring Hamiltonicity

• Authors: Bo Ning
• Venue: Information Processing Letters
• Year: 2013

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@?V(H), d"H(u,v)=2 implies that max{d(u),d(v)}=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 R of graphs, G is R-f-heavy if G is R-f-heavy for every R@?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.