Some Semi-equivelar Maps of Euler Characteristics-2
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Some Semi-equivelar Maps of Euler Characteristics-2 Debashis Bhowmik1 • Ashish Kumar Upadhyay1
Received: 27 May 2020 / Revised: 1 October 2020 / Accepted: 7 October 2020 Ó The National Academy of Sciences, India 2020
Abstract Semi-equivelar maps are generalizations of Archimedean solids. We classify all the semi-equivelar maps on the surface of Euler Characteristics-2 with vertices up to 12. We calculate their automorphism groups and study their vertex-transitivity. Keywords Semi-equivelar maps Archimedean solids
We consider surface as compact, connected, 2-manifold without boundary. By a map, we mean a polyhedral map on a surface. Equivelar and semi-equivelar maps (SEM) are generalizations of Platonic and Archimedean solids, see [1] for details. An automorphism of a map is a permutation on the vertex set that preserves incidences. A map K is vertextransitive if its automorphism group AutðKÞ acts transitively on the vertex set VðKÞ. A SEM is denoted by ½an11 ; . . .; anl l (see e.g. [2]) . For a SEM K, link of a vertex x 2 VðKÞ is the subcomplex fr 2 K : x 62 r&r [fxg 2 Kg. We associate a graph Gi ðKÞ to each SEM K as follows: the vertex set is VðGi ðKÞÞ ¼ VðKÞ, and the edges are ½x; y 2 EGðGi ðKÞÞ if jNðxÞ \ NðyÞj ¼ i, where x; y 2 VðKÞ, and NðxÞ ¼ fx1 2 VðKÞ : x1 2 VðlkðxÞÞg, see [3]. In this article, K will denote a SEM of Euler Characteristics-2.
& Ashish Kumar Upadhyay [email protected] Debashis Bhowmik [email protected] 1
Department of Mathematics, Indian Institute of Technology Patna, Bihta, Bihar 801103, India
The objective of this article is to present complete classification of K with at most 12 vertices. We begin with following known result: Proposition 1 [3, 4] Let K be of type ½an11 ; . . .; anl l . If ½an11 ; . . .; anl l ¼ ½37 ; ½35 ; 41 , ½34 ; 42 , ½31 ; 44 , ½32 ; 41 ; 31 ; 61 , ½34 ; 81 , ½31 ; 41 ; 81 ; 41 , ½31 ; 61 ; 41 ; 61 , ½43 ; 61 , ½41 ; 61 ; 161 , ½41 ; 81 ; 121 or ½62 ; 81 , then there exists a vertex-transitive map of respective type. Moreover, there are 34 SEM of type ½37 out of which 6 are orientable. From Euler characteristic equation, it is clear that there does not exist any K with vertices less than or equal to 11, and for 12 vertices, all the possible types of maps on this surface are ½32 ; 42 ; 61 , ½33 ; 62 , ½42 ; 62 , ½37 , ½34 ; 42 , ½33 ; 41 ; 31 ; 41 and ½31 ; 44 . It is easy to see that 6 9 SEMs corresponding to map types ½32 ; 42 ; 61 , ½33 ; 62 , ½42 ; 62 , and the map of type ½37 has been classified in [3]. In this article, we show: Theorem 1
Let K be with 12 vertices.
T1 If K is of type ½34 ; 42 then it is isomorphic to KO1½34 ;42 ; KO2½34 ;42 or KNO½34 ;42 , see 1. The maps KO1½34 ;42 and KO2½34 ;42 are orientable and KNO½34 ;42 is non-orientable. T2 If K is of type ½31 ; 44 then it is isomorphic to KNO1½31 ;44 , KNO2½31 ;44 or KO1½31 ;44 , see 1. KNO1½31 ;44 , KNO2½31 ;44 are non-orientable and KNO2½31 ;44 is orientable. T3 If K is of type ½33 ; 41 ; 31 ; 41 th
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