Oriented diameter of graphs with diameter 3

In 1978, Chvátal and Thomassen proved that every 2-edge-connected graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value f ( d ) such that every 2-edge-connected graph G with diameter d has an orientation with diameter at most f ( d ) . For d = 3 , their general bounds reduce to 8 ⩽ f ( 3 ) ⩽ 24 . We improve these bounds to 9 ⩽ f ( 3 ) ⩽ 11 .