Partial domination in supercubic graphs

Abstract

For some $\alpha$ with $0<\alpha \le 1$, a subset $X$ of vertices in a graph $G$ of order~$n$ is an $\alpha$-partial dominating set of $G$ if the set $X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial domination number ${\mathrm{pd}}_{\alpha }(G)$ of $G$ is the minimum cardinality of an $\alpha$-partial dominating set of $G$. In this paper partial domination of graphs with minimum degree at least $3$ is studied. It is proved that if $G$ is a graph of order~$n$ and with $\delta (G)\ge 3$, then ${\mathrm{pd}}_{\frac{7}{8}}(G)\le \frac{1}{3}n$. If in addition $n\ge 60$, then ${\mathrm{pd}}_{\frac{9}{10}}(G)\le \frac{1}{3}n$, and if $G$ is a connected cubic graph of order $n\ge 28$, then ${\mathrm{pd}}_{\frac{13}{14}}(G)\le \frac{1}{3}n$. Along the way it is shown that there are exactly four connected cubic graphs of order $14$ with domination number $5$.