Claw-free graphs and 2-factors that separate independent vertices

  • Authors:

  • Ralph J. Faudree;Colton Magnant;Kenta Ozeki;Kiyoshi Yoshimoto

  • Affiliations:

  • Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee38152;Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania18015;National Institute of Informatics, Tokyo, 101-8430, Japan;Department of Mathematics, Nihon University, Tokyo, 101-8308, Japan

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

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Abstract

In this article, we prove that a line graph with minimum degree δ≥7 has a spanning subgraph in which every component is a clique of order at least three. This implies that if G is a line graph with δ≥7, then for any independent set S there is a 2-factor of G such that each cycle contains at most one vertex of S. This supports the conjecture that δ≥5 is sufficient to imply the existence of such a 2-factor in the larger class of claw-free graphs. It is also shown that if G is a claw-free graph of order n and independence number α with δ≥2n/α−2 and n≥3α3/2, then for any maximum independent set S, G has a 2-factor with α cycles such that each cycle contains one vertex of S. This is in support of a conjecture that δ≥n/α≥5 is sufficient to imply the existence of a 2-factor with α cycles, each containing one vertex of a maximum independent set. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 251–263, 2012, © 2012 Wiley Periodicals, Inc. (Kenta Ozeki is a Research Fellow of the Japan Society for the Promotion of Science.)