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Domination in a graph with a 2‐factor
Author(s) -
Kawarabayashi Kenichi,
Plummer Michael D.,
Saito Akira
Publication year - 2006
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20142
Subject(s) - combinatorics , mathematics , cubic graph , domination analysis , graph , upper and lower bounds , discrete mathematics , girth (graph theory) , symmetric graph , line graph , voltage graph , vertex (graph theory) , mathematical analysis
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 3 k 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