On 2-regular graphs whose girth is one less than the maximum

We say that a digraph is 2-regular (dichotomous) if the out-degrees d0(j) and in-degrees d1(j) of any its vertex j ε V satisfy the equality d0(j) = d1(j) = 2. A graph Γ is said to be primitive if for any pair i and j of its vertices in Γ there exists a path from i to j of length m > 0. The least m is denoted γ(Γ) and called the exponent of Γ. Let G(n, 2, p) stand for the class of strongly connected 2-regular graphs with n vertices of girth (the length of the shortest circuit) p, and let P(n, 2, p) denote the class of primitive 2-regular graphs of girth p with n vertices. The girth of a 2-regular graph with n vertices does not exceed ]n/2[, where ]x[ is the least integer no smaller than x. Earlier, the author proved that any primitive 2-regular graph with n vertices and with the maximal possible girth ]n/2[ had the exponent equal exactly to n - 1. In this paper we prove that for odd n greater than or equal 13 G(n, 2, (n - 1)/2) = P(n, 2, (n - 1)/2), any graph in G(n, 2, (n - 1)/2) has a circuit of length (n + 1)/2, and for any Γ ε G(n, 2, (n - 1)/2) the inequality γ(Γ) [less-than or equal to] (n - 1)2/4 + 5 is true.