Domination in 4-Regular Graphs with Girth 3

In this paper, we investigate the domination number, independent domination number, connected domination number, total domination number denoted by γ(Gn),γi(Gn),γc(Gn,γt(Gn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma (G\left( n \right)), \gamma_{i} (G\left( n \right)), \gamma_{c} (G\left( n \right),\gamma_{t} (G\left( n \right)) $$\end{document} respectively for 4-regular graphs of n vertices with girth 3. Here, G(n) denotes the 4-regular graphs of n vertices with girth 3. We obtain some exact values of G(n) for these parameters. We further establish that γiGn=γGnforn≥6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma_{i} \left( {G\left( n \right)} \right) = \gamma \left( {G\left( n \right)} \right)\, {\text{for }} n \ge 6 $$\end{document} and γcGn=γtGn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma_{c} \left(G\left( n \right)\right) = \gamma_{t} \left( {G\left( n \right)} \right) $$\end{document} for n ≥ 6. Nordhaus–Gaddum type results are also obtained for these parameters.