Implicit degree condition for hamiltonicity of 2-heavy graphs

Let id(v) denote the implicit degree of a vertex v in a graph G. An induced subgraph S of G is called f-implicit-heavy if max{id(x),id(y)}≥|V(G)|/2 for every pair of vertices x,y∈V(S) at distance 2 in S. For a given graph R, G is called R-f-implicit-heavy if every induced subgraph of G isomorphic to R is f-implicit-heavy. For a family R of graphs, G is called R-f-implicit-heavy if G is R-f-implicit-heavy for every R∈R. G is called 2-heavy if there are at least two end-vertices of every induced claw (K1,3) in G have degree at least |V(G)|/2. In this paper, we prove that: Let G be a 2-connected 2-heavy graph. If G is {P7,D}-f-implicit-heavy or {P7,H}-f-implicit-heavy, then G is hamiltonian.