Decomposition of circulant digraphs with two jumps into cycles of equal lengths

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Abstract

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.

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APA

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

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