Large Girth and Small Oriented Diameter Graphs

Abstract

In 2015, Dankelmann and Bau proved that for every bridgeless graph G of order n and minimum degree δ there is an orientation of diameter at most 11 n δ+1 + 9. In 2016, Surmacs reduced this bound to 7 n δ+1 . In this paper, we consider the girth of a graph g and show that for any ε > 0 there is a bound of the form (2g + ε) n h(δ,g) + O(1), where h(δ, g) is a polynomial. Letting g = 3 and ε < 1 gives an inprovement on the result by Surmacs.