Abstract
LetHbe a graph onnvertices and𝒢a collection ofnsubgraphs ofH, one for each vertex, where𝒢is an orthogonal double cover (ODC) ofHif every edge ofHoccurs in exactly two members of𝒢and any two members share an edge whenever the corresponding vertices are adjacent inHand share no edges whenever the corresponding vertices are nonadjacent inH. In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by new disjoint unions of complete bipartite graphs.
Highlights
For the definition of an orthogonal double cover (ODC) of the complete graph Kn by a graph G and for a survey on this topic, see [1]
In [2], this concept has been generalized to ODCs of any graph H by a graph G
If all the pages are isomorphic to a given graph G, G is said to be an ODC of Kn,n by G
Summary
For the definition of an orthogonal double cover (ODC) of the complete graph Kn by a graph G and for a survey on this topic, see [1]. An algebraic construction of ODCs via “symmetric starters” (see Section 2) has been exploited to get a complete classification of ODCs of Kn,n by G for n ≤ 9, a few exceptions apart, all graphs G are found this way (see [3, Table 1]). (iii) All 3-regular Cayley graphs on Abelian groups, except K4 and the 3-prism (Cartesian product of C3 and K2), have ODCs by 3K2. (ii) All 3-regular Cayley graphs on Abelian groups, except K4, have ODCs by P3 ∪ K2. Much research on this subject focused on the detection of ODCs with pages isomorphic to a given graph G. The other terminologies not defined here can be found in [6]
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