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

• By restricting Fan-type degree condition to induced subgraphs of graphs, we enlarge the graph classes ensuring Hamiltonicity.
• It is the first time to find certain triples of Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs.
• Our work will give new views to the topic of Hamiltonicity, and stimulate the further studies for finding sufficient conditions for Hamiltonicity.

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 Fanʼs 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)u,v∈V(H), dH(u,v)=2dH(u,v)=2 implies that max{d(u),d(v)}⩾n/2max{d(u),d(v)}⩾n/2. For a given graph R, G   is called R-fR-f-heavy if every induced subgraph of G isomorphic to R is f  -heavy. For a family RR of graphs, G   is R-fR-f-heavy if G is R-f  -heavy for every R∈RR∈R. In this note we show that every 2-connected graph G has a Hamilton cycle if G   is {K1,3,P7,D}{K1,3,P7,D}-f  -heavy or {K1,3,P7,H}{K1,3,P7,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.