Korean J. Math.  Vol 20, No 4 (2012)  pp.
DOI: https://doi.org/10.11568/kjm.2012.20.4.

EVERY LINK IS A BOUNDARY OF A COMPLETE BIPARTITE GRAPH $K_{2,n}$

Yongjun Jang, Snag-Min Jeon, Dongseok Kim

Abstract


A voltage assignment on a graph was used to enumerate all possible 2-cell embeddings of a graph onto surfaces. The boundary of the surface which is obtained from 0 voltage on every edges of a very special diagram of a complete bipartite graph $K_{m,n}$ is surprisingly the $(m, n)$ torus link. In the present article, we prove
that every link is the boundary of a complete bipartite multi-graph
$K_{m,n}$ for which voltage assignments are either −1 or 1 and that every link is the boundary of a complete bipartite graph $K_{2,n} for which voltage assignments are either −1, 0 or 1 where edges in the diagram of graphs may be linked but not knotted.


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ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr