Regular embeddings of Kn,n where n is a power of 2.: I:: Metacyclic case

被引:40
作者
Du, Shao-Fei [1 ]
Jones, Gareth
Kwak, Jin Ho
Nedela, Roman
Skoviera, Martin
机构
[1] Capital Normal Univ, Dept Math, Beijing 100037, Peoples R China
[2] Univ Southampton, Sch Math, Southampton SO17 1BJ, Hants, England
[3] Pohang Univ Sci & Technol, Combinatorial & Computat Math Ctr, Pohang 790784, South Korea
[4] Slovak Acad Sci, Math Inst, Banska Bystrica 97549, Slovakia
[5] Comenius Univ, Fac Math Phys & Informat, Bratislava 84248, Slovakia
基金
中国国家自然科学基金;
关键词
D O I
10.1016/j.ejc.2006.08.012
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs K-n.n, where n = 2(e). The method involves groups G which factorize as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of K-n.n, based on that for G. We prove that given n = 2(e) (for e >= 3), there are, up to map isomorphism, exactly 2(e-2) + 4 regular embeddings of K-n.n. Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not. (C) 2006 Elsevier Ltd. All rights reserved.
引用
收藏
页码:1595 / 1609
页数:15
相关论文
共 28 条
[1]  
[Anonymous], [No title captured]
[2]  
[Anonymous], 1953, MATH Z
[3]  
Biggs N., 1971, Journal of Combinatorial Theory, Series B, V11, P132, DOI 10.1016/0095-8956(71)90023-2
[4]  
BIGGS NL, 1979, LONDN MATH SOC LECT, V33
[5]  
Coxeter H. S. M., 1972, GENERATORS RELATIONS
[6]  
DOUGLAS J, 1961, P NATL ACAD SCI USA, V49, P1493
[7]   Regular embeddings of complete multipartite graphs [J].
Du, SF ;
Kwak, JH ;
Nedela, R .
EUROPEAN JOURNAL OF COMBINATORICS, 2005, 26 (3-4) :505-519
[8]   A classification of regular embeddings of graphs of order a product of two primes [J].
Du, SF ;
Kwak, JH ;
Nedela, R .
JOURNAL OF ALGEBRAIC COMBINATORICS, 2004, 19 (02) :123-141
[9]  
DU SF, IN PRESS DISCRETE MA
[10]  
DU SF, UNPUB NONMETACYCLIC