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Graphs and Combinatorics

, Volume 32, Issue 2, pp 453–461 | Cite as

Spectrum of Mixed Bi-uniform Hypergraphs

  • Maria Axenovich
  • Enrica Cherubini
  • Torsten Ueckerdt
Original Paper
  • 97 Downloads

Abstract

A mixed hypergraph is a triple \(H=(V,{\mathcal {C}},{\mathcal {D}})\), where V is a set of vertices, \({\mathcal {C}}\) and \({\mathcal {D}}\) are sets of hyperedges. A vertex-coloring of H is proper if C-edges are not totally multicolored and D-edges are not monochromatic. The feasible set S(H) of H is the set of all integers, s, such that H has a proper coloring with s colors. Bujtás and Tuza (Graphs Combin 24:1–12, 2008) gave a characterization of feasible sets for mixed hypergraphs with all C- and D-edges of the same size \(r, r\ge 3\). In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all C-edges of size \(\ell \) and all D-edges of size m, where \(\ell , m \ge 2\). Moreover, we show that for every sequence \((r(s))_{s=\ell }^n, n \ge \ell \), of natural numbers there exists such a hypergraph with exactly r(s) proper colorings using s colors, \(s = \ell ,\ldots ,n\), and no proper coloring with more than n colors. Choosing \(\ell = m=r\) this answers a question of Bujtás and Tuza, and generalizes their result with a shorter proof.

Keywords

Mixed hypergraph Vertex coloring Spectrum Feasible set 

Mathematics Subject Classification

05C15 05C65 05C30 

Notes

Acknowledgments

The authors thank the anonymous referee for a careful reading and spotting minor errors.

References

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Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Maria Axenovich
    • 1
  • Enrica Cherubini
    • 1
  • Torsten Ueckerdt
    • 1
  1. 1.Karlsruher Institut für TechnologieKarlsruheGermany

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