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

  • Authors:

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

  • Affiliations:

  • Department of Public Finance and Taxation, Takming University of Science and Technology, Taipei City 11451, Taiwan, ROC;Intel-NTU Connected Context Computing Center, National Taiwan University, Taipei 10617, Taiwan, ROC;Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li City 32023, Taiwan, ROC

  • Venue:
  • Information Processing Letters
  • Year:
  • 2012

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Abstract

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@?V, unless G belongs to one of the two exceptional families of graphs, denoted by G"1 and G"2. Moreover, G-{e} is hamiltonian for any e@?E, unless G is one of the two particular types of graphs in G"1.