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Bulletin of Pure and Applied Sciences- Math& Stat. (Started in 1982)
eISSN: 2320-3226
pISSN: 0970-6577
Impact Factor: 4.895 (2017)
DOI: 10.5958/2320-3226
Editor-in-Chief:  Prof. Dr. Lalit Mohan Upadhyaya
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Article Details

Bulletin of Pure and Applied Sciences- Math& Stat. (Started in 1982)
Year : 2020, Volume & Issue : BPAS-Maths & Stat 39E(1), JAN-JUN 2020
Page No. : 115-121, Article Type : Original Aticle
Article DOI : 10.5958/2320-3226.2020.00011.9 (Communicated, edited and typeset in Latex by Lalit Mohan Upadhyaya (Editor-in-Chief). Received May 03, 2019 / Revised January 24, 2020 / Accepted February 17, 2020. Online First Published on June 30, 2020)

Nonoverlapping partitions of a surface

Simon Davis1
Author’s Affiliation : 1. Research Foundation of Southern California, 8861 Villa La Jolla Drive #13595, La Jolla, CA 92039, U.S.A. 1. E-mail: [email protected]

Corresponding Author : Simon Davis,
E-Mail:-[email protected]


Abstract

The colouring of planar domains is considered through the tight packing of rectangular regions.It is demonstrated that a maximal number of colours in a neighbourhood is achieved through the introduction of ribboned regions. This number can be reduced to four in the brick model with a special choice of colours in the surrounding region. An exceptional planar domain found by interweaving a ribboned region with a compact hexagonal configuration of isometric circles of a Schottky group requires an additional colour. The equivalent tight packing of isometric circles of the Schottky group provides a method for deriving the number of colours required to cover a Riemann surface.  The chromatic number is derived for both orientable surfaces of genus $gge 3$ and nonorientable surfaces of genus $gge 4$.

Keywords

planar domains, ribboned regions, isometric circles, minimal number, Schottky problem, Riemann surfaces. 2020 Mathematics Subject Classification : 05C15, 14H42, 14H55, 20H10, 30C20, 30FXX, 30F50, 30F99, 52C15, 52C17, 52C26, 52C99. PACS: 55A15, 14H55, 30F40.
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