On a 2-factor with a specified edge in a graph satisfying the Ore condition

  • Authors:

  • Atsushi Kaneko;Kiyoshi Yoshimoto

  • Affiliations:

  • Department of Computer Science and Communication Engineering, Kogakuin University, 1-24-2 Nishi-Shinjuku, Shinjuku-ku, Tokyo 163-8677, Japan;Department of Mathematics, College of Science and Technology, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308 Japan

  • Venue:
  • Discrete Mathematics - Kleitman and combinatorics: a celebration
  • Year:
  • 2002

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Abstract

Let k be an integer greater than one, and let G be a simple graph with at least 4k + 1 vertices. In this article, we prove that if σ2(G) ≥ |V(G)|, then for an edge e of G, there exists a 2-factor with k cycles that contains e, or |V(G)| is even and G has a vertex cover of size |V(G)|/2 containing the endpoints of e. Here σ2(G) is the minimum degree sum for a pair of nonadjacent vertices.