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The Graph of Critical Pairs of a Crown

  • Fidel Barrera-Cruz
  • Rebecca Garcia
  • Pamela Harris
  • Bethany Kubik
  • Heather SmithEmail author
  • Shannon Talbott
  • Libby Taylor
  • William T. Trotter
Article
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Abstract

There is a natural way to associate with a poset P a hypergraph , called the hypergraph of critical pairs, so that the dimension of P is exactly equal to the chromatic number of . The edges of have variable sizes, but it is of interest to consider the graph G formed by the edges of that have size 2. The chromatic number of G is less than or equal to the dimension of P and the difference between the two values can be arbitrarily large. Nevertheless, there are important instances where the two parameters are the same, and we study one of these in this paper. Our focus is on a family \(\{{S_{n}^{k}}:n\ge 3, k\ge 0\}\) of height two posets called crowns. We show that the chromatic number of the graph \({G_{n}^{k}}\) of critical pairs of the crown \({S_{n}^{k}}\) is the same as the dimension of \({S_{n}^{k}}\), which is known to be ⌈2(n + k)/(k + 2)⌉. In fact, this theorem follows as an immediate corollary to the stronger result: The independence number of \({G_{n}^{k}}\) is (k + 1)(k + 2)/2. We obtain this theorem as part of a comprehensive analysis of independent sets in \({G_{n}^{k}}\) including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.

Keywords

Dimension Chromatic number Critical pairs 

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Notes

Acknowledgments

The research reported here was initiated in two informal workshops hosted by the United States Military Academy (West Point) in 2015 and 2016, and we thank them for providing a stimulating and encouraging atmosphere.

Smith acknowledges support from NSF-DMS grant #1344199. Harris acknowledges support from NSF-DMS grant #1620202.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.SunnyvaleUSA
  2. 2.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA
  3. 3.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  4. 4.Department of Mathematics and StatisticsUniversity of Minnesota DuluthDuluthUSA
  5. 5.Department of Mathematics and Computer ScienceDavidson CollegeDavidsonUSA
  6. 6.Department of Mathematics and Computer ScienceMoravian CollegeBethlehemUSA
  7. 7.Department of MathematicsStanford UniversityStanfordUSA
  8. 8.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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