Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]

Year 2016, Volume: 3 Issue: 2, 105 - 123, 15.05.2016

Daniel Khoshnoudirad

https://doi.org/10.13069/jacodesmath.00924

Abstract

The equation (4) on the page 178 of the paper previously published has to be corrected. We had only handled the case of the Farey vertices for which
$\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*}$.
In fact we had to distinguish two cases: $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*}$ and $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)=0$.
However, we highlight the correct results of the original paper and its applications.
We underline that in this work, we still brought several contributions. These contributions are: applying the fundamental formulas of Graph Theory to the Farey diagram of order $(m,n)$, finding a good upper bound for the degree of a Farey vertex and the relations between the Farey diagrams and the linear diophantine equations.

Keywords

Combinatorial number theory, Farey diagrams, Theoretical computer sciences, Discrete planes, Diophantine equations, Arithmetical geometry, Combinatorial geometry, Discrete geometry, Graph theory in computer sciences

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