The maximum valency of regular graphs with given order and odd girth

The odd girth of a graph G is the length of a shortest odd cycle in G . Let d ( n, g ) denote the largest k such that there exists a k ‐regular graph of order n and odd girth g . It is shown that d n, g ≥ 2| n / g ≥ if n ≥ 2 g . As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2 j ‐regular graph of order n and odd girth g if and only if n ≥ gj , where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.