Odd complete minors in even embeddings on surfaces

Fijavž, Gašper; Nakamoto, Atsuhiro

doi:10.1016/j.disc.2015.07.015

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…Therefore, if H is an odd minor of G, then G and H have the same cycle parity. By Theorem 3 and the argument in [3], we can prove the following corollary. Corollary 4.…”

Section: Corollarymentioning

confidence: 84%

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Even Embeddings of the Complete Graphs and Their Cycle Parities

Noguchi

2016

Journal of Graph Theory

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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.

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“…Therefore, if H is an odd minor of G, then G and H have the same cycle parity. By Theorem 3 and the argument in [3], we can prove the following corollary. Corollary 4.…”

Section: Corollarymentioning

confidence: 84%

“…Therefore, if H is an odd minor of G , then G and H have the same cycle parity. By Theorem and the argument in , we can prove the following corollary. Corollary Every locally planar nonbipartite even embedding on a closed surface F 2 has an odd minor that is a complete quadrangulation if and only if F 2 admits a complete quadrangulation.…”

Section: Applicationmentioning

confidence: 88%

Even Embeddings of the Complete Graphs and Their Cycle Parities

Noguchi

2016

Journal of Graph Theory

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“…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.…”

Section: Preliminarymentioning

confidence: 99%