Connectivity of graphs with given girth pair

Girth pairs were introduced by Harary and Kovacs [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g,h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g,h) such that g is odd and h>=g+3 is even has high (vertex-)connectivity if its diameter is at most h-3. The edge version of all results is also studied.