Local exponents of primitive two-colored digraph with cycles of length s and 2s − 1

Abstract

Let D be a digraph on n vertices {v1, v2,…, vn}. A two-colored digraph D(2) is a digraph D such that each of its arcs is colored by red or blue. An (h, k)-walk is a walk with precisely h red arcs and k blue arcs. A strongly connected two-colored digraph is primitive if there are nonnegative integers h and k such that for each two vertices vi and vj there is a (h, k)-walk from vi to vj and a (h, k)-walk from vj to vi. The local exponent of a two-colored digraph D(2) at the vertex vt, denoted by exp(vt, D(2)), is the least positive integer ht + kt over all nonnegative integers ht and kt such that for every vertex vi, i = 1, 2,…, vn there is a (ht, kt)-walk from vi to vt. For some positive integer s ≥ 6, we discuss the local exponent of each vertex vt in D(2) where D(2) is a primitive two-colored digraph contains precisely a cycle of length s and a cycle of length 2s − 1.