Common extremal graphs for three inequalities involving domination parameters

Document Type: Research Paper

Author

University of Architecture, Civil Engineering and Geodesy (UACEG)

Abstract

‎Let $\delta (G)$‎, ‎$\Delta (G)$ and $\gamma(G)$‎ ‎be the minimum degree‎, ‎maximum degree and‎ ‎domination number of a graph $G=(V(G)‎, ‎E(G))$‎, ‎respectively‎. ‎A partition of $V(G)$‎, ‎all of whose classes are dominating sets in $G$‎, ‎is called a domatic partition of $G$‎. ‎The maximum number of classes of‎ ‎a domatic partition of $G$ is called the domatic number of $G$‎, ‎denoted $d(G)$‎. ‎It is well known that‎ ‎$d(G) \leq \delta(G)‎ + ‎1$‎, ‎$d(G)\gamma(G) \leq |V(G)|$ \cite{ch}‎, ‎and $|V(G)| \leq (\Delta(G)‎+‎1)\gamma(G)$ \cite{berge}‎. ‎In this paper‎, ‎we investigate the graphs $G$ for which‎ ‎all the above inequalities become simultaneously equalities‎.

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