Even Embeddings of the Complete Graphs and Their Cycle Parities

Abstract

The complete graph Kn on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F2 with Euler characteristic εfalse(F2false)=nfalse(5−nfalse)/4 if and only if n≡0,5(mod8) (resp. n≡0,1(mod4) and n≠5). In this article, we shall show that if Kn quadrangulates a closed surface F2, then Kn has a quadrangular embedding on F2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism ρ:π1false(F2false)→double-struckZ2, called the cycle parity.

Spanning bipartite quadrangulations of triangulations of the projective plane

“…A quadrangulation of Σ is a special case of an evenly embedded graph, that is, one such that each face is bounded by a closed walk of even length. The graph is also called even embedding, for example, [6,18]. Since bipartite graphs have no cycles of odd length, any bipartite graph should be evenly embedded on any surface.…”