Partial domination in supercubic graphs

Abstract

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