Cyclic orthogonal double covers of 4-regular circulant graphs

An orthogonal double cover (ODC) of a graph H is a collection G={Gv: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 Gu and Gv in G, |E(Gu)∩E(Gv)| 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.

► 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.