A note on domination, girth and minimum degree

Let image be a graph of order image, minimum degree image, girth image and domination number image. 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 image. 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 image determined a finite set of graphs image such that image unless image is a cycle or image. Our main result is that for every image there is a finite set of graphs image such that image unless image is a cycle or image. Furthermore, we conjecture another improvement of Brigham and Duttonʹs bound and prove a weakened version of this conjecture.