Cyclic orthogonal double covers of 4-regular circulant graphs

An orthogonal double cover (ODC) of a graph H is a collection G = { G v : v � V ( H ) } of | V ( H ) | subgraphs of H such that every edge of H is contained in exactly two members of G and for any two members G u and G v in G , | E ( G u ) � E ( G v ) | is 1 if u and v are adjacent in H and it is 0 if u and v are nonadjacent in H . An ODC G of H is cyclic (CODC) if the cyclic group of order | V ( H ) | is a subgroup of the automorphism group of G . In this paper, we are concerned with CODCs of 4-regular circulant graphs. Highlights� Let G be a simple graph with four edges and let H be a 4-regular circulant graph.�Problem: Does there exists a cyclic orthogonal double cover of H by G ? � In this study, we have completely settled this problem. � We use a special kind of labelling for its proof.