On the 1-fault hamiltonicity for graphs satisfying Oreʼs theorem

On the 2-Vertex Fault Hamiltonicity for Graphs satisfying Ore's Theorem

“…The graphs in ( 4) and (6) given in Theorem 5 above need more explanations. For G ∈ K h : w : K t in (4), K t is a graph by removing some (none, one, or more) vertex-disjoint edges of K t , with h ≤ t; in the operation of ":", edges are added from w to K h and K t as long as σ Case 1.…”

“…Following the previous study in [6], we are interested in the fault-tolerance of graphs satifying the degree conditions given by Ore [3]. Since the 1-fault tolerance has been thoroughly studied in [6], we further explore the 2-fault tolerance for any graph G with σ 2 (G) ≥ n, where |G| = n. This paper has given a conclusion that any G with σ 2 (G) ≥ n and κ(G) ≥ 4 must be 2-vertex-fault tolerant unless G belongs to one of the nine graph families in Theorem 6. Whether a given graph G under the same…”

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For any undirected and simple graph G = (V, E), where V denotes the vertex set and E the edge set of G. G is called hamiltonian if it contains a cycle that visits each vertex of G exactly once. Ore (1960) ≥ n holds for every nonadjacent pair of vertices u and v in V , where n is the total number of distinct vertices of G. Kao et al (2012) proved that any graph G satisfying Ore's theorem remains hamiltonian after the removal of any vertex x ∈ V unless G belongs to one of the two exceptional families of graphs. In this paper, we proved that in fact, any graph satisfying Ore's theorem remains hamiltonian after the removal of two vertices x, y ∈ V unless G belongs to one of the nine exceptional families of graphs.

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

“…k-edge-fault hamiltonian, k-edge-fault hamiltonian-connected, and k-edge-fault r * -connected) graph. It is proved in [16] that if the condition d G (u) + d G (v) ≥ n holds for any nonadjacent pair of vertices {u, v} in a simple graph G = (V , E) with |V | = n ≥ 3, then G is 1-fault hamiltonian unless it belongs to two exceptional families. To our knowledge, this is the first paper which establishes the sufficient condition for a general graph to be fault-tolerant hamiltonian.…”

“…In [16], we introduced another two families of graphs and obtained the 1-fault version of Theorem 1 for k = 0. Let G 1,n and G 2,n be two graph families consisting of n-vertexed graphs such that Figure 1 for an illustration.)…”