A Note on Singular Edges and Hamiltonicity in Claw-Free Graphs with Locally Disconnected Vertices

Abstract

An edge e of a graph G is called singular if it is not on a triangle; otherwise, e is nonsingular. A vertex is called singular if it is adjacent to a singular edge; otherwise, it is called nonsingular. We prove the following. Let G be a connected claw-free graph such that every locally disconnected vertex x ∈ V(G) satisfies the following conditions: Then G is either hamiltonian, or G is the line graph of the graph obtained from $$K_{2,3}$$ by attaching a pendant edge to its each vertex of degree two. Some results on forbidden subgraph conditions for hamiltonicity in 3-connected claw-free graphs are also obtained as immediate corollaries