Oriented diameter of graphs with given maximum degree

In this article, we show that every bridgeless graph G of order n and maximum degree has an orientation of diameter at most n-+3. We then use this result and the definition NG(H)=?vV(H)NG(v)\V(H), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high-degree vertices, namely: For every edge e, G has an orientation of diameter at most n-|NG(e)|+4, if e is on a triangle and at most n-|NG(e)|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n-|NG(H)|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|- deg G(s))+7, for every partite set A of G and sV(G)\A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n-2+7. For each bound, we give a polynomial-time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.