<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers <italic>S</italic> whose elements do not differ by an element of a given set <italic>M</italic> of positive integers. We find some exact values and some bounds for the maximal density when the elements of <italic>M</italic> are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order <italic>r</italic> is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with <italic>r</italic> predetermined terms and each term afterwards is the sum of <italic>r</italic> preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order <italic>r</italic>. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set <italic>M</italic>.</p></abstract>

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

Motzkin’s Maximal Density and Related Chromatic Numbers

Department of Mathematics, Indian Institute of Technology Patna

Department of Mathematics, Indian Institute of Technology Roorkee

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Motzkin\u2019s Maximal Density and Related Chromatic Numbers"}

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an...

Motzkin’s Maximal Density and Related Chromatic Numbers

Publié en ligne: 20 juil. 2018

Pages: 27 - 45

Reçu: 10 nov. 2016

Accepté: 23 mars 2017

DOI: https://doi.org/10.1515/udt-2018-0002

Mots clés
maximal density, generalized Fibonacci numbers, fractional coloring, circular coloring

© 2018 Anshika Srivastava et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Motzkin’s Maximal Density and Related Chromatic Numbers

Publié en ligne: 20 juil. 2018

Pages: 27 - 45

Reçu: 10 nov. 2016

Accepté: 23 mars 2017

DOI: https://doi.org/10.1515/udt-2018-0002

Mots clésmaximal density, generalized Fibonacci numbers, fractional coloring, circular coloring

© 2018 Anshika Srivastava et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
maximal density, generalized Fibonacci numbers, fractional coloring, circular coloring