A note on domination, girth and minimum degree

Let GG be a graph of order nn, minimum degree δ⩾2δ⩾2, girth g⩾5g⩾5 and domination number γγ. 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⌉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}i∈{1,2} determined a finite set of graphs GiGi such that γ⩽⌈n/2-g/6-(3i+3)/6⌉γ⩽⌈n/2-g/6-(3i+3)/6⌉ unless GG is a cycle or G∈GiG∈Gi.Our main result is that for every i∈Ni∈N there is a finite set of graphs GiGi such that γ⩽n/2-g/6-iγ⩽n/2-g/6-i unless GG is a cycle or G∈GiG∈Gi. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.