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Graphs and Closed Surfaces Associated with a Pairing of Edges for Regular Polygons

Graphs and Closed Surfaces Associated with a Pairing of Edges for Regular Polygons In this paper, we define the concept of graph extension, embedded on a closed and orientable surface, associated to a pairing of edges of regular polygons in order to show that the K -regular pairing of edges graphs can be obtained by the canonical extension of graphs (graphs with a single vertex). We will present examples of K -regular graphs associated to surfaces with genus g ≤ 3. Keywords Trivalent graphs · Closed surfaces · Pairing of edges · Surgeries Mathematics Subject Classification 14J80 · 57M15 · 57N05 1 Introduction Given a polygon P with 2E sides, always is possible to obtain a closed orientable surface M by edge pairing (quotient map), where the image of the border of P corresponds to a graph G with E edges embedded on M (see Fig. 1). Some authors have searched graphs that can be associated with an edge pairing and possible edge pairings linked to each of them. Jorgensen and Naatanen (1982) showed that for E = 9, there are eight trivalent pairings (all vertices have degree 3) for surfaces with genus g = 2. These pairings are associated with five non-isomorphic graphs (see Fig. 12). For g = 3, Nakamura http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Graphs and Closed Surfaces Associated with a Pairing of Edges for Regular Polygons

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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-019-00163-y
Publisher site
See Article on Publisher Site

Abstract

In this paper, we define the concept of graph extension, embedded on a closed and orientable surface, associated to a pairing of edges of regular polygons in order to show that the K -regular pairing of edges graphs can be obtained by the canonical extension of graphs (graphs with a single vertex). We will present examples of K -regular graphs associated to surfaces with genus g ≤ 3. Keywords Trivalent graphs · Closed surfaces · Pairing of edges · Surgeries Mathematics Subject Classification 14J80 · 57M15 · 57N05 1 Introduction Given a polygon P with 2E sides, always is possible to obtain a closed orientable surface M by edge pairing (quotient map), where the image of the border of P corresponds to a graph G with E edges embedded on M (see Fig. 1). Some authors have searched graphs that can be associated with an edge pairing and possible edge pairings linked to each of them. Jorgensen and Naatanen (1982) showed that for E = 9, there are eight trivalent pairings (all vertices have degree 3) for surfaces with genus g = 2. These pairings are associated with five non-isomorphic graphs (see Fig. 12). For g = 3, Nakamura

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Jul 31, 2019

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