Balloons, cut‐edges, matchings, and total domination in regular graphs of odd degree
O Suil, Douglas B. West · Journal of Graph Theory · 2009
Abstract Aballoonin a graphGis a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge ofG. Letb(G) be the number of balloons, letc(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}${\mathcal{F}}_{{{n}},{{r}}}$\end{document} be the family of connected (2r+1)‐regular graphs withnvertices, and let\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$\end{document} . For\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$\end{document} , we prove the sharp inequalitiesc(G)⩽[r(n−2)−2]/(2r2+2r−1)−1 and α′(G)⩾n/2−rb/(2r+1). Usingb⩽[(2r−1)n+2]/(4r2+4r−2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and eachr, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γt(G) of a cubic graph, we prove γt(G)⩽n/2−b(G)/2 (except that γt(G) may ben/2−1 whenb(G)=3 and the balloons cover all but one vertex). With α′(G)⩾n/2−b(G)/3 for cubic graphs, this improves the known inequality γt(G)⩽α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010