Complete bipartite multi-graphs with a unique regular dessin

Chen, Jiyong1; Fan, Wenwen2

2021-02-01

10.1007/s10801-021-01019-9

JOURNAL OF ALGEBRAIC COMBINATORICS   影响因子和分区

0925-9899

1572-9192

A regular dessin is an orientable edge-regular bipartite map. Jones et al.(J Combin Theory Ser B 98:241-248, 2008) showed that a complete bipartite graph K-n,K-n has a unique orientably (arc-)regular map if and only if gcd(n, phi(n)) = 1. We extended this result in Fan and Li (J Graph Theory 87:581-586, 2018) by proving that a complete bipartite graph K-m,K-n underlies a unique regular dessin if and only if gcd(m, phi(n)) = 1 and gcd(n, phi(m)) = 1. In this paper, it is shown that a complete bipartite multi-graph K-m,n((lambda)) with lambda > 1 underlies a unique regular dessin if and only if lambda = 2, gcd(m(2), n(2)) = 1, and gcd(m, phi(n(2 '))) gcd(n, phi(m(2 '))) = 1, where n(2) and n(2 ') denote the 2-part and 2 '-part of n, respectively. Furthermore, apart from two degenerated cases, each of such dessins can be uniquely decomposed into the direct product of two dessins such that one is symmetric and reflexible, and the other has only one face.

Dessins Regular Complete bipartite multi-graph Orientable Unique

SCI

英语

第一

NSFC[12061092,11501497,11931005] ; Yunnan Applied Basic Research Project[2015FD013]

Mathematics

Mathematics

WOS:000617428900001

SPRINGER

MATHEMATICS

Web of Science

1.Southern Univ Sci & Technol, Dept Math, Shenzhen, Guangdong, Peoples R China
2.Yunnan Normal Univ, Sch Math, Kunming, Yunnan, Peoples R China

Chen, Jiyong,Fan, Wenwen. Complete bipartite multi-graphs with a unique regular dessin[J]. JOURNAL OF ALGEBRAIC COMBINATORICS,2021.

Chen, Jiyong,&Fan, Wenwen.(2021).Complete bipartite multi-graphs with a unique regular dessin.JOURNAL OF ALGEBRAIC COMBINATORICS.

Chen, Jiyong,et al."Complete bipartite multi-graphs with a unique regular dessin".JOURNAL OF ALGEBRAIC COMBINATORICS (2021).