Abstract
An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graphKmn,mn(for any values ofm,n) and all graphs in the collection are isomorphic to certain spanning subgraphs. Furthermore, the ODCs ofKn,nby certain disjoint stars are constructed.
Highlights
Let Ᏻ be a collection of n spanning subgraphs of the complete graph on n vertices
An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs
If all pages in Ᏻ are isomorphic to a graph G, Ᏻ is said to be an ODC by G
Summary
Let Ᏻ be a collection of n spanning subgraphs (called pages) of the complete graph on n vertices. Vn−1) be a symmetric starter of an ODC of Kn,n by G with respect to Zn. If we define the graph G to be a spanning subgraph of Kmn,mn such that E(G ) = {{(vi, 0), (vi + i + αn, 1)} : i ∈ Zn, α ∈ Zm, vi + i + αn ∈ Zmn}, we can deduce the following theorem.
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