Edge-transitive uniface embeddings of bipartite multi-graphs
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Edge-transitive uniface embeddings of bipartite multi-graphs Wenwen Fan1
· Cai Heng Li2 · Naer Wang3
Received: 20 October 2017 / Accepted: 8 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract It is shown that a bipartite multi-graph Γ has an orientably edge-transitive (λ) such that gcd(m, n) = 1 and embedding with a single face if and only if Γ = Km,n mn is even whenever λ is even. A consequence of this shows that each orientable surface carries a simple edge-transitive map with a single face. Keywords Edge-transitive · Uniface embedding · Complete bipartite multi-graphs Mathematics Subject Classification 05C25 · 54C20 · 54C25 · 57M10
This work was partially supported by NSFC: 11501497, 11771200; Yunnan Applied Basic Research Projects: 2015FD013, and Natural Science Foundation of Zhejiang Province: LY16A010010, LQ17A010003.
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Wenwen Fan [email protected] Cai Heng Li [email protected] Naer Wang [email protected]
1
School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, People’s Republic of China
2
Department of Mathematics, Southern University of Science and Technology, ShenZhen, People’s Republic of China
3
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316000, Zhejiang, People’s Republic of China
123
J Algebr Comb
1 Introduction Let Γ = (V, E) be a connected (multi)graph with vertex set V and edge set E. A 2-cell embedding of Γ in an orientable closed surface S decomposes S into open discs, called faces. The set of faces is denoted by F. A 2-cell embedding is sometimes simply called an embedding, or a map, and denoted by an incident triple M = (V, E, F). The graph Γ is called the underlying graph, and the surface S is called the supporting surface of the map. A map is said to be a simple map if the underlying graph is a simple graph, while a map called a bipartite map if it has a bipartite underlying graph. If M has only one face, then M is called a uniface map or a uniface embedding. An automorphism of a map M is a permutation of V ∪ E ∪ F which preserves V , E, and F (setwise) and the incidence relation between them. For a bipartite map M = (V, E, F), the vertex set V has a bipartition V = U ∪ W such that all edges of M lie between U and W . Let Aut⊕ M be the subgroup of the automorphisms of M which preserves the orientation of the supporting surface and the bipartition of V . Then Aut ⊕ M is semiregular on the edge set E. If Aut⊕ M is transitive on E, then M is called orientably edge-transitive map (or embedding). In this case, Aut ⊕ M is regular on E, and M is called an orientably edge-regular map (or embedding). Classifying edge-transitive embeddings of complete bipartite (multi)graphs is an open problem proposed by Jones in 2007, see [8]. Some special families have been classified, refer to [2–5,7]. In terms of [9], the maps we consider in this paper are of (λ) type 2ex. We denote by Km,n a complete bipartite (multi)graph with edge-multiplicity λ. The following theorem characterises orientably
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