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A two-colored digraph D(2) is primitive provided there are nonnegative integers h and k such that for each pair of not necessarily distinct vertices u and v in D(2) there exists a (h, k)-walk in D(2) from u to v. The exponent of a primitive twocolored digraph D(2), exp(D(2)), is the smallest positive integer h+k taken over all such nonnegative integers h and k. The exponent of a vertex v in D(2) is the smallest positive integer s+ t such that for each vertex u in D(2) there is an (s, t)-walk from v to u. We study the vertex exponents of primitive two-colored digraphs L (2) n on n ≥ 5 vertices whose underlying digraph is the Hamiltonian digraph consisting of the cycle v1 → vn → vn−1 → · · · → v2 → v1 and the arc v1 → vn−2. For such two-colored digraph it is known that 2n2 − 6n+ 2 ≤ exp(L n ) ≤ (n3 − 2n2 + 1)/2. We show that if exp(L (2) n ) = (n 3 − 2n2 + 1)/2, then its vertex exponents lie on [(n3 − 2n2 − 3n+4)/4, (n3 − 2n2 +3n+6)/4] and if exp(L n ) = 2n2 − 6n+2, then its vertex exponents lie on [n2 − 4n+ 5, n2 − 2n− 1].

VERTEX EXPONENTS OF A CLASS OF TWO-COLORED HAMILTONIAN DIGRAPHS