On the Hamiltonian-Connectedness for Graphs Satisfying Ore’s Theorem

Shih, Yuan-Kang; Su, Hsun; Kao, Shin-Shin

doi:10.1007/978-3-642-35452-6_4

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The 1-fault Hamiltonicity Of Ore's Theorem1

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2014

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“…To derive the 1-fault hamiltonian-connected under the condition d G (u) + d G (v) ≥ n + 1 in Theorem 1, we first notice the close relation between k-fault hamiltonian graphs and (k − 1)-fault hamiltonianconnected graphs [6]. The following theorem was derived by Zhao [18] (with certain errors) and the authors [15] independently. It is interesting to note that besides K 3 , the exceptional families in Theorem 6 are the same as those in Theorem 5.…”

Section: The 1-fault Hamiltonicity Of Ore's Theoremmentioning

confidence: 95%

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

Shih

Kao

2014

International Journal of Computer Mathematics

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Consider any undirected and simple graph G = (V , E), where V and E denote the vertex set and the edge set of G, respectively. Let |G| = |V | = n. The well-known Ore's theorem states that if deg G (u) + deg G (v) ≥ n + k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k = −1, hamiltonian for k = 0, and hamiltonian-connected for k = 1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r * -connected for 1 ≤ r ≤ k + 2 with k ≥ 1. In this paper, we improve both theorems by showing that the hamiltonicity or r * -connectivity of any graph G satisfying the condition deg G (u) + deg G (v) ≥ n + k with k ≥ −1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.

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Section: The 1-fault Hamiltonicity Of Ore's Theoremmentioning

confidence: 95%