Abstract
Dankelmann, Guo and Surmacs proved that every bridgeless graph G of order n with given maximum degree Δ(G) has an orientation of diameter at most n−Δ(G)+3 [J. Graph Theory, 88(1)(2018), 5-17]. They also constructed a family of bridgeless graphs whose oriented diameter reaches this upper bound. In this paper, we show that G has an orientation of diameter at most n−⌊g(G)−12⌋(Δ(G)−4)−1, where g(G) is the girth of G. Moreover, we construct several families of bridgeless graphs whose oriented diameter attains n−⌊g(G)−12⌋(Δ(G)−4)−1, and prove that the upper bound is tight for Δ(G)≥4. We also give a necessary condition for a bridgeless graph to attain this upper bound. Furthermore, we verify that if G is a 3-connected graph with girth at least 5, then the oriented diameter of such G is at most n−⌊g(G)−12⌋(Δ(G)−4)−2.
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