The competition index of two-colored non-hamiltonian digraph with two cycles whose lengths differ by three

Abstract

A two-colored digraph is a digraph each of whose arcs is colored by red or blue. An (h, g)-walk in a two-colored digraph is a walk consisting of h red arcs and g blue arcs. A strongly connected two-colored digraph is primitive provided there exists a positive integer h + g such that for each ordered pair of vertices x and y there is a (h, g)-walk from x to y. The competition index of a primitive two-colored digraph is the least positive integer h + g over all pairs of nonnegative integers h and g such that for each pair of vertices x and y in there is a vertex w such that there are (h, g)-walks from x to w and from y to w. For some positive integer s ≥ 7 (s ≢ 0 mod 3), we discuss the competition index of a class of strongly connected primitive two-colored digraph D(2) consisting of two cycles of lengths s and s + 3, respectively. We present the competition index of such primitive two-colored digraph D(2) in terms of s and the lengths of two specific paths in D(2).