A lower bound on the order of regular graphs with given girth pair

The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovacs [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209-218]. A (delta, g)-cage is a smallest delta-regular graph with girth g. For all delta >= 3 and odd girth g >= 5, Harary and Kovbcs conjectured the existence of a (delta, g)-cage that contains a cycle of length g + 1. In the main theorem of this article we present a lower bound on the order of a delta-regular graph with odd girth g >= 5 and even girth h >= g + 3. We use this bound to show that every (delta, g)-cage with delta >= 3 and g is an element of {5, 7} contains a cycle of length g + 1, a result that can be seen as an extension of the aforementioned conjecture by Harary and Kovacs for these values of delta, g. Moreover, for every odd g >= 5, we prove that the even girth of all (delta, g)-cages with delta large enough is at most (3g - 3)/2. (c) 2007 Wiley Periodicals, Inc.