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Distance Graphs Generated by Five Primes (Research)

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Advances in Mathematical Sciences

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 21))

Abstract

Let D be a set of prime numbers. We investigate the chromatic number of graphs with vertex set of the integers and edges between any pair of vertices whose absolute difference falls in D. Such a graph is called a prime distance graph, and the set D is called the distance set. The chromatic number of prime distance graphs is known when the distance set D has at most four primes. In this paper we begin to classify prime distance graphs with a distance set of five primes. In particular, we completely classify the family of distance sets D = {2, 3, 7, 19, p} where p is any prime, and solve most of the more general family D = {2, 3, 7, p, q} for any primes p and q. The number theoretic function κ(D) is used as a tool, and some general properties about κ(D) are established.

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Acknowledgements

The research is partially supported by the National Science Foundation under grant DMS-1600778 and the National Aeronautics and Space Administration under grant NASA MIRO NX15AQ06A.

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Authors and Affiliations

  1. California State University, Los Angeles, CA, USA

    Daphne Der-Fen Liu & Angel Chavez

  2. University of Washington, Seattle, WA, USA

    Grant Robinson

Authors
  1. Daphne Der-Fen Liu
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  2. Grant Robinson
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  3. Angel Chavez
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Corresponding author

Correspondence to Daphne Der-Fen Liu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Bahar Acu

  2. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Donatella Danielli

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

    Marta Lewicka

  4. Department of Mathematics, St. Thomas University, Houston, TX, USA

    Arati Pati

  5. Security and Privacy, Bosch Research and Technology Center, Pittsburgh, PA, USA

    Saraswathy RV

  6. Department of Mathematics, Lehigh University, Bethlehem, PA, USA

    Miranda Teboh-Ewungkem

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Liu, D.DF., Robinson, G., Chavez, A. (2020). Distance Graphs Generated by Five Primes (Research). In: Acu, B., Danielli, D., Lewicka, M., Pati, A., Saraswathy RV, Teboh-Ewungkem, M. (eds) Advances in Mathematical Sciences. Association for Women in Mathematics Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-42687-3_3

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