Domination in Graphs of Minimum Degree at least Two and Large Girth

We prove that for graphs of order n, minimum degree 2 and girth g 5 the domination number satisfies 1 3 + 2 3g n. As a corollary this implies that for cubic graphs of order n and girth g 5 the domination number satisfies 44 135 + 82 135g n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic graphs, manuscript (2005)) and Kawarabayashi, Plummer and Saito (Domination in a graph with a 2-factor, J. Graph Theory 52 (2006), 1-6) for large enough girth. Furthermore, it confirms a conjecture due to Reed about connected cubic graphs (Paths, stars and the number three, Combin. Prob. Comput. 5 (1996), 267-276) for girth at least 83. Keywords domination number; minimum degree; girth; cubic graph

• Type: Journal
• Year: 2008
• Where: GC
• Authors: Christian Löwenstein, Dieter Rautenbach
• Keywords: Applied Computing, Cubic Graph, Domination Number Satisfies, GC 2008, Minimum Degree