Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws†

Abstract

Let G = (V(G), E(G)) be a connected graph. The distance between two vertices x and y in G, denoted by dG (x, y), is the length of a shortest path between x and y. A graph G is called almost distance-hereditary, if each connected induced subgraph H of G has the property that dH (u, v) ≤ dG (u, v) + 1 for every pair of vertices u and v in H. A graph G is 2-heavy if the degree of at least two end vertices of each claw in G are greater than or equal to |V(G)|/2. We show that every 2-connected, 2-heavy and almost distance-hereditary graph has a Hamiltonian cycle. This generalizes the main result of [8] that states: every 2-connected, claw-free and almost distance-hereditary graph is Hamiltonian. †Dedicated to Prof. Dr Dr h.c. Hubertus Th. Jongen on his 60th birthday.