Regular embeddings of Kn, n where n is an odd prime power
Regular embeddings of Kn, n where n is an odd prime power
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DOI：
10.1016/j.ejc.2005.07.021
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发表时间：
2007-08
期刊：
影响因子：
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通讯作者：
G. Jones;R. Nedela;M. Škoviera
中科院分区：
文献类型：
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作者：
G. Jones;R. Nedela;M. Škoviera
证明了如果n= pe其中p是奇素数且e≥1，则完全二部图Kn，n在可定向曲面中有pe− 1正则嵌入.这些映射是循环群和二面体群的凯莱映射，有{2n，n}型和亏格（n−1）（n−2）/2;一个是自反的，其余的是手征的。该方法涉及的群体分解为产品的两个循环群的顺序n。我们推出，如果n是奇数，则Kn，n至少有n/n/p| np可定向的正则嵌入，并且这个下界是可达到的当且仅当没有两个素数p和q除n满足p <$1mod（q）。
We show that if n=pewhere p is an odd prime and e≥1, then the complete bipartite graph Kn,nhas pe−1regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n,n} and genus (n−1)(n−2)/2; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n. We deduce that if n is odd then Kn,nhas at least n/∏p|np orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p≡1mod(q).