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Biased Expansions of Biased Graphs and Their Chromatic Polynomials

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Abstract

A biased graph is a graph with a distinguished set of circles, such that if two circles in the set are contained in a theta graph, then so is the third circle of the theta graph. We introduce a new biased graph, a biased expansion of a biased graph, that satisfies certain lifting and projection properties with the original biased graph. We relate the chromatic polynomials of a biased graph and its biased expansions, thus generalizing a biased-graph result of Zaslavsky [7] and a hyperplane result of Ehrenborg and Readdy [1]. We also determine which biased expansions have supersolvable bias matroids.

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References

  1. Ehrenborg R., Readdy M.A.: The Dowling transform of subspace arrangements. J. Combin. Theory Ser. A 91(1-2), 322–333 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Koban L.: Comments on “Supersolvable frame-matroid and graphic-lift lattices” by T. Zaslavsky [European Journal of Combinatorics 22 (2001) 119]. European J. Combin. 25(1), 141–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Orlik P.: Introduction to Arrangements. Amer. Math. Soc., Providence, RI (1989)

    Google Scholar 

  4. Stanley R.P.: Supersolvable lattices. Algebra Universalis 2, 197–217 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zaslavsky, T.: The Möbius function and the characteristic polynomial. In: White, N. (ed.) Combinatorial Geometries, pp. 114–138. Cambridge Univ. Press, Cambridge (1987)

  6. Zaslavsky T.: Biased graphs. I. bias, balance, and gains. J. Combin. Theory Ser. B 47(1), 32–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zaslavsky T.: Biased graphs. III. chromatic and dichromatic invariants. J. Combin. Theory Ser. B 64(1), 17–88 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zaslavsky T.: Supersolvable frame-matroid and graphic-lift lattices. European J. Combin. 22(1), 119–133 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zaslavsky, T.: Biased graphs. V. group and biased expansions. In preparation

  10. Zaslavsky T.: Associativity in multiary quasigroups: the way of biased expansions. Aequationes Math. 83(1-2), 1–66 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics and Computer Science, University of Maine at Farmington, 228 Main St., Farmington, ME, 04938, USA

    Lori Koban

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  1. Lori Koban
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Correspondence to Lori Koban.

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Koban, L. Biased Expansions of Biased Graphs and Their Chromatic Polynomials. Ann. Comb. 16, 781–788 (2012). https://doi.org/10.1007/s00026-012-0160-7

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  • DOI: https://doi.org/10.1007/s00026-012-0160-7

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