Let G=Gn(a1,a2) be a connected circulant digraph of order n with two distinct jumps a1,a2<n. We give several sufficient conditions for a decomposition of Gn(a1,a2) into directed cycles of equal lengths. We then prove that Gn(a1,a2) contains a 2-factor consisting of all cycles of equal lengths and comprised of both jumps if and only if gcd(n,s1a1+s2a2)=k(s1+s2) and a1≡a2(mods1+s2) for some positive integers k,s1,s2. Based on this last result we prove that Gn(a1,a2) can be decomposed into two 2-factors with all cycles comprising both jumps if and only if gcd(n,s1a1+s2a2)=gcd(n,s1a2+s2a1)=k(s1+s2) and a1≡a2(mods1+s2) for some positive integers k,s1,s2. Furthermore, we prove that if such a decomposition exists then all resulting cycles are of equal lengths.
Bogdanowicz, Z. R. (2015). Decomposition of circulant digraphs with two jumps into cycles of equal lengths. Discrete Applied Mathematics, 180, 45–51. https://doi.org/10.1016/j.dam.2014.08.007