Fan-type implicit-heavy subgraphs for hamiltonicity of implicit claw-heavy graphs

• We show that an implicit claw-heavy graph is hamiltonian by imposing implicit degree conditions on some subgraphs.
• Our results extend a previous theorem of Ning [B. Ning, Inf. Process. Lett. 113 (2013) 823–826].
• We give an example to show that our result is stronger than Ning's result.

Let G be a graph on n   vertices and id(v)id(v) denote the implicit degree of a vertex v in G. We call G implicit claw-heavy if every induced claw of G has a pair of nonadjacent vertices such that their implicit degree sum is at least n. And we call an induced subgraph S of G implicit f  -heavy if max⁡{id(u),id(v)}≥n/2max⁡{id(u),id(v)}≥n/2 for every pair of vertices u,v∈V(S)u,v∈V(S) at distance 2 in S. For a given graph R, G is called implicit R-f-heavy if every induced subgraph of G isomorphic to R is implicit f  -heavy. For a family RR of graphs, G   is called implicit RR-f-heavy if G is implicit R-f  -heavy for every R∈RR∈R. In this paper, we prove that: Let G be a 2-connected implicit claw-heavy graph. If G   is implicit {P7,D}{P7,D}-f  -heavy or implicit {P7,H}{P7,H}-f-heavy, then G is hamiltonian.