Dense bipartite circulants and their routing via rectangular twisted torus

Whereas the maximum number of vertices in a four-regular circulant of diameter a is 2a^2+2a+1, the bound turns out to be 2a^2 if the condition of bipartiteness is imposed. Tzvieli presented a family of @j(a) such dense bipartite circulants on 2a^2 vertices for each a>=3, where @j(a) denotes the number of positive integers less than or equal to @?12(a-1)@? that are coprime with a. The present paper shows that each of those graphs is obtainable from the 2axa rectangular twisted torus by appropriately trading a maximum of 2a edges for as many new edges. The underlying structural similarity between the two graphs leads to a simple intuitive routing algorithm for the circulants. The result closely parallels the routing in dense nonbipartite circulants on 2a^2+2a+1 vertices. Additional results include a set of vertex-disjoint paths between every pair of distinct vertices in the circulants, and a proof that the 2axa rectangular twisted torus itself is probably not a circulant.

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