Balloons, cut-edges, matchings, and total domination in regular graphs of odd degree

• Authors: Suil O;Douglas B. West
• Venue: Journal of Graph Theory
• Year: 2010

A balloon in a graph G is a maximal 2-edge-connected subgraph incident to exactly one cut-edge of G. Let b(G) be the number of balloons, let c(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 with n vertices, 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 inequalities c(G)⩽[r(n-2)-2]-(2r2+2r-1)-1 and α′(G)⩾n-2-rb-(2r+1). Using b⩽[(2r-1)n+2]-(4r2+4r-2), we obtain a simple proof of the bound \documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*}\alpha\prime(G)\ge\frac{n}{2}-\frac{r}{2}\frac{(2r-1)n+2}{(2r+1)(2r^2+2r-1)}\end{eqnarray*}\end{document}proved by Henning and Yeo. For each of these bounds and each r, 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 be n-2-1 when b(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