Hamilton-connected indices of graphs

Document Type

Article

Publication Date

July 2009

Publication Title

Discrete Mathematics

First Page

4819

Last Page

4827

DOI

https://doi.org/10.1016/j.disc.2008.06.030

Abstract

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131–148] defined hc(G) to be the least integer m such that the iterated line graph Lm(G) is Hamilton-connected. Let diam(G) be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G. In this paper, we show that k−1≤hc(G)≤max{diam(G),k−1}. We also show that κ3(G)≤hc(G)≤κ3(G)+2 where κ3(G) is the least integer m such that Lm(G) is 3-connected. Finally we prove that hc(G)≤|V(G)|−Δ(G)+1. These bounds are all sharp.

Notes

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