Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 5 (2015), 825-831.

The chromatic polynomials of signed Petersen graphs

Matthias Beck, Erika Meza, Bryan Nevarez, Alana Shine, and Michael Young

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Abstract

Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they can be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at any positive integer. We developed and executed a computer program (running in SAGE) that efficiently determines the number of proper k-colorings for a given signed graph; our computations for the signed Petersen graphs confirm Zaslavsky’s conjecture. We also computed the chromatic polynomials of all signed complete graphs with up to five vertices.

Article information

Source
Involve, Volume 8, Number 5 (2015), 825-831.

Dates
Received: 18 April 2014
Revised: 18 December 2014
Accepted: 13 January 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511370951

Digital Object Identifier
doi:10.2140/involve.2015.8.825

Mathematical Reviews number (MathSciNet)
MR3404660

Zentralblatt MATH identifier
1322.05073

Subjects
Primary: 05C22: Signed and weighted graphs
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05C15: Coloring of graphs and hypergraphs

Keywords
signed graph Petersen graph complete graph chromatic polynomial zero-free chromatic polynomial

Citation

Beck, Matthias; Meza, Erika; Nevarez, Bryan; Shine, Alana; Young, Michael. The chromatic polynomials of signed Petersen graphs. Involve 8 (2015), no. 5, 825--831. doi:10.2140/involve.2015.8.825. https://projecteuclid.org/euclid.involve/1511370951


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References

  • W. A. Stein et al., “Sage mathematics software”, 2012, hook http://www.sagemath.org \posturlhook. Version 5.1.
  • T. Zaslavsky, “Signed graph coloring”, Discrete Math. 39:2 (1982), 215–228.
  • T. Zaslavsky, “Signed graphs”, Discrete Appl. Math. 4:1 (1982), 47–74. Erratum in 5:2 (1983), p. 248.
  • T. Zaslavsky, “A mathematical bibliography of signed and gain graphs and allied areas”, Electron. J. Combin./Dyn. Surv. 8 (1998–2012).
  • T. Zaslavsky, “Six signed Petersen graphs, and their automorphisms”, Discrete Math. 312:9 (2012), 1558–1583.

Supplemental materials

  • Sage code for computing chromatic polynomials of signed graphs.