Domination in a graph with a 2‐factor
Ken‐ichi Kawarabayashi, Michael D. Plummer, Akira Saito · Journal of Graph Theory · 2006
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