Implicit heavy subgraph conditions for hamiltonicity of almost distance-hereditary graphs

We recently proved two new results on the hamiltonicity of almost distance-hereditary graphs, involving implicit degree conditions on claws, i.e., induced subgraphs isomorphic to K_1,3. A graph G of order n is called implicit 1-heavy if at least one of the end vertices of each induced claw of G has implicit degree at least n/2, and G is called implicit claw-heavy if each induced claw of G has a pair of end vertices with implicit degree sum at least n. A graph G is said to be almost distance-hereditary if each connected induced subgraph H of G has the property d_H(x, y) = d_G(x, y) + 1 for any pair of vertices x, y ? V (H). We recently proved that every 2-connected implicit claw-heavy almost distance-hereditary graph is hamiltonian, and that every 3-connected implicit 1-heavy almost distance-hereditary graph is hamiltonian. These results improve two recent results due to Chen and Ning.