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

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: (i)if x is nonsingular of degree 4, then x is on an induced cycle of length at least 4 with at most 4 nonsingular edges,(ii)if x is not nonsingular of degree 4, then x is on an induced cycle of length at least 4 with at most 3 nonsingular edges,(iii)if x is of degree 2, then x is singular and x is on an induced cycle C of length at least 4 with at most 2 nonsingular edges such that G[ V(C) ∩ V₂(G) ] is a path or a cycle.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.