A note on domination, girth and minimum degree

Abstract

Let G be a graph of order n , minimum degree δ ⩾ 2 , girth g ⩾ 5 and domination number γ . In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269–275] proved that γ ⩽ ⌈ n / 2 - g / 6 ⌉ . This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131–141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195–212] who for i ∈ { 1 , 2 } determined a finite set of graphs G i such that γ ⩽ ⌈ n / 2 - g / 6 - ( 3 i + 3 ) / 6 ⌉ unless G is a cycle or G ∈ G i . Our main result is that for every i ∈ N there is a finite set of graphs G i such that γ ⩽ n / 2 - g / 6 - i unless G is a cycle or G ∈ G i . Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.