Closures, cycles, and paths

  • Authors:

  • Jochen Harant;Arnfried Kemnitz;Akira Saito;Ingo Schiermeyer

  • Affiliations:

  • Institut für mathematiktechnische universität ilmenau, 98 684 ilmenau, Germany;Computational mathematics technische universität braunschweig 38 023 braunschweig, Germany;Department of computer science nihon university, sakurajosui 3-25-40 setagaya-ku, Tokyo, 156-8550, Japan;Institut für diskrete mathematik und algebratechnische universität bergakademie freiberg09 596 Freiberg, Germany

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

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Abstract

In 1960 Ore proved the following theorem: Let G be a graph of order n. If d(u) + d(v)≥n for every pair of nonadjacent vertices u and v, then G is hamiltonian. Since then for several other graph properties similar sufficient degree conditions have been obtained, so-called “Ore-type degree conditions”. In [R. J. Faudree, R. H. Schelp, A. Saito, and I. Schiermeyer, Discrete Math 307 (2007), 873–877], Faudree et al. strengthened Ore's theorem as follows: They determined the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Ore's condition) but still imply that the graph is hamiltonian. In this article we prove that for some other graph properties the corresponding Ore-type degree conditions can be strengthened as well. These graph properties include traceable graphs, hamiltonian-connected graphs, k-leaf-connected graphs, pancyclic graphs, and graphs having a 2-factor with two components. Graph closures are computed to show these results. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 314–323, 2012 © 2012 Wiley Periodicals, Inc.