An upper bound on the domination number of -vertex connected cubic graphs

In 1996, Reed proved that the domination number γ ( G ) of every n -vertex graph G with minimum degree at least 3 is at most 3 n / 8 . This bound is sharp for cubic graphs if there is no restriction on connectivity. In this paper we show that γ ( G ) ≤ 4 n / 11 for every n -vertex cubic connected graph G if n > 8 . Note that Reed’s conjecture that γ ( G ) ≤ ⌈ n / 3 ⌉ for every connected cubic n -vertex graph G is not true.