Local exponents of a certain class of two-colored Hamiltonian digraphs with two cycles whose lengths n and n - 3

Abstract

A two-colored digraph is a digraph for which each of its arc is colored by red or blue. An (h, ℓ)-walk is a walk consisting of h red arcs and ℓ blue arcs. A strongly connected two-colored digraph is primitive provided there exist nonnegative integers h and ℓ such that for each ordered pair of vertices u and v there is a(h, ℓ)-walk from u to v. The local exponent of a vertex v in a primitive two-colored digraph D (2), denoted by exp(v, D (2)) is the least positive integer h + ℓ over all nonnegative integers h and ℓ such that for vertex u in D (2) there is an (h, ℓ)-walk from u to v. We discuss local exponent of primitive two-colored Hamiltonian digraph on n vertices consisting of two cycles of lengths n and n - 3, respectively. For each vertex v in , we present an explicit formula for exp(v,D3(2)).