Domination in a graph with a 2-factor

  • Authors:

  • Ken-ichi Kawarabayashi;Michael D. Plummer;Akira Saito

  • Affiliations:

  • Graduate School of Information Sciences, Tohoku University, Aramaki aza Aoba 09, Aoba-ku, Sendai, Miyagi 980-8579, Japan;Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240;Department of Computer Science, Nihon University, Sakurajosui 3–25–40, Setagaya-ku, Tokyo 156–8550, Japan

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2006

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Abstract

Let γ(G) be the domination number of a graph G. Reed [6] proved that every graph G of minimum degree at least three satisfies γ(G) ≤ (3-8)|G|, and conjectured that a better upper bound can be obtained for cubic graphs. In this paper, we prove that a 2-edge-connected cubic graph G of girth at least 3k satisfies $\gamma (G)\le (({3k+2}))/({9k+3})|G|$. For $k\ge 3$, this gives $\gamma (G)\le ({11}/{30})|G|$, which is better than Reed's bound. In order to obtain this bound, we actually prove a more general theorem for graphs with a 2-factor. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 1–6, 2006 The study is dedicated to Professor Hikoe Enomoto on his sixtieth birthday.