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Febuary 2020 Combinatorics of $k$-Farey graphs
Jonah Gaster, Miguel Lopez, Emily Rexer, Zoë Riell, Yang Xiao
Rocky Mountain J. Math. 50(1): 135-151 (Febuary 2020). DOI: 10.1216/rmj.2020.50.135

Abstract

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the k-Farey graphs k and k, two natural variants of the Farey graph in which we relax the edge condition to indicate intersection number =k or k, respectively.

The former, k, is disconnected when k>1. In fact, we find that the number of connected components is infinite if and only if k is not a prime power. Moreover, we find that each component of k is a quasitree (in fact, a tree when k is even) and Aut(k) is uncountable for k>1.

As for k, Agol obtained an upper bound of 1+ min{p:p is a prime>k} for both chromatic and clique numbers, and observed that this is an equality when k is either one or two less than a prime. We add to this list the values of k that are three less than a prime equivalent to 11(mod12), and we show computer-assisted computations of many values of k for which equality fails.

Citation

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Jonah Gaster. Miguel Lopez. Emily Rexer. Zoë Riell. Yang Xiao. "Combinatorics of $k$-Farey graphs." Rocky Mountain J. Math. 50 (1) 135 - 151, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.135

Information

Received: 31 May 2019; Revised: 17 July 2019; Accepted: 19 July 2019; Published: Febuary 2020
First available in Project Euclid: 30 April 2020

zbMATH: 07201558
MathSciNet: MR4092548
Digital Object Identifier: 10.1216/rmj.2020.50.135

Subjects:
Primary: 05C63, 57M15, 57M60

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 1 • Febuary 2020
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