A new bound on the domination number of connected cubic graphs

In 1996, Reed proved that the domination number, $\gamma(G)$, of every $n$-vertex graph $G$ with minimum degree at least $3$ is at most $3n/8$. This bound is sharp for cubic graphs if there is no restriction on connectivity. In this paper, improving an upper bound by Kostochka and Stodolsky we show that for $n>8$ the domination number of every $n$-vertex cubic connected graph is at most $\lfloor 5n/14\rfloor$. This bound is sharp for even $8<n\leq18$.