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Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph

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Abstract

Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L k(G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ≥ k + 1 ≥ 4, then in G k, for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T,  h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G,  h(G) ≤ hc(G) ≤ max{h(G) + 1, k′ + 1}, where k′ is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Eminjan Sabir & Elkin Vumar

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  1. Eminjan Sabir
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  2. Elkin Vumar
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Correspondence to Elkin Vumar.

Additional information

This work is supported by NSFC (No. 11061034) and XJEDU2010I01.

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Sabir, E., Vumar, E. Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph. Graphs and Combinatorics 30, 1551–1563 (2014). https://doi.org/10.1007/s00373-013-1362-4

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