On the exponents of two-colored digraphs with two cycles†

Abstract

We consider the special primitive two-colored digraphs whose uncolored digraph has n þ s vertices and consists of one n-cycle and one ðn À tÞ-cycle. We give some primitivity conditions and an upper bound on the exponent. Further, for the case s ¼ 0, we give a tight upper bound on the exponent and the characterization of extremal two-colored digraphs.