Local exponents of two-coloured bi-cycles whose lengths differ by 1

A two-coloured digraph (2) d is a digraph each of whose arc is coloured by red or blue. an (h,k)-walk in a two-coloured digraph is a walk of length (h,k) consisting of h red arcs and k blue arcs. a two-coloured digraph (2) d is primitive provided that for each pair of vertices u and v there exists an (h,k)-walk from u to v. the inner local exponent of a vertex v in (2) d,denoted as (2) expin(v,d ),is the smallest positive integer h+k over all nonnegative integers h and k such that for each vertex u in (2) d there is an (h,k)-walk from u to v. we study the inner local exponent of primitive two-coloured digraphs consisting of exactly two cycles of length s+1 and s,respectively. let 0 u be the vertex of indegree 2 in d(2) . for each vertex v in d(2),we show that expin(v,d(2)) = expin(u0,d(2)) + d(u0,v) where d(u0,v) is the distance from u0 to v.