Let G be a bridgeless graph. We introduce the maximum edge girth of G, denoted by g^* (G) =\lG (e) e E (G) \, where lG (e) is the edge girth of e, defined as the length of the shortest cycle containing e. Let F (d, A) be the smallest value for which every bridgeless graph G with diameter d and g^* (G) A admits a strong orientation G such that the diameter of G is at most F (d, A). Let f (d) =F (d, A), where A=\a N 2 a 2d+1\. Chvátal and Thomassen (JCT-B, 1978) obtained general bounds for f (d) and showed that f (2) =6. Kwok et al. (JCT-B, 2010) proved that 9 f (3) 11. Wang and Chen (JCT-B, 2022) determined f (3) =9. In this paper, we give that 12 F (4, A^*) 13, where A^*=\2, 3, 6, 7, 8, 9\.
Lin et al. (Thu,) studied this question.
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