Elsevier

Discrete Mathematics

Volume 338, Issue 11, 6 November 2015, Pages 2042-2050
Discrete Mathematics

Hamiltonian claw-free graphs with locally disconnected vertices

https://doi.org/10.1016/j.disc.2015.04.020Get rights and content
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Abstract

An edge of G is singular if it does not lie on any triangle of G; otherwise, it is non-singular. A vertex u of a graph G is called locally connected if the induced subgraph G[N(u)] by its neighborhood is connected; otherwise, it is called locally disconnected.

In this paper, we prove that if a connected claw-free graph G of order at least three satisfies the following two conditions: For each locally disconnected vertex v of G with degree at least 3, there is a nonnegative integer s such that v lies on an induced cycle of length at least 4 with at most s non-singular edges and with at least s3 locally connected vertices; for each locally disconnected vertex v of G with degree 2, there is a nonnegative integer s such that v lies on an induced cycle C with at most s non-singular edges and with at least s2 locally connected vertices and such that the subgraph induced by those vertices of C that have degree two in G is a path or a cycle, then G is Hamiltonian, and it is best possible in some sense.

Our result is a common extension of two known results in Bielak (2000) and in Li (2002); hence also of the results in Oberly and Sumner (1979) and in Ryjáček (1990).

Keywords

Claw-free graph
Hamiltonian
Closure
Locally disconnected vertex
Singular edge

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