On the 1-fault hamiltonicity for graphs satisfying Ore's theorem
Consider 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. It is proved by Ore that G is hamiltonian if deg(G)(u)+deg(G)(v) >= n holds for every nonadjacent pair of vertices u and v in V. where n is the total number of distinct vertices of G. In this paper, we prove that in fact G - {x}, is hamiltonian for any x is an element of V. unless G belongs to one of the two exceptional families of graphs, denoted by G(i) and G(2). Moreover, G - {e} is hamiltonian for any e is an element of E, unless G is one of the two particular types of graphs in G(1). (C) 2012 Elsevier B.V. All rights reserved.
ore's theorem hamiltonian combinatorial problems degree 1-fault hamiltonian