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We say that a digraph is 2-regular (dichotomous) if the out-degrees d 0 (j) and in-degrees d 1 (j) of any its vertex j ∈ V satisfy the equality d 0 (j) = d 1 (j) = 2. A graph Г is said to be primitive if for any pair i and j of its vertices in Г there exists a path from i to j of length m &gt; 0. The least m is denoted y(Г) and called the exponent of Г. 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.

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