Minimum choosability of planar graphs

  • Huijuan Wang
  • Bin Liu
  • Ling Gai
  • Hongwei Du
  • Jianliang Wu
Article
  • 13 Downloads

Abstract

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring \(\phi \) such that \(\phi (x) \in L(x)\). Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring \(\phi \) such that \(\phi (x) \in L(x)\). We proved \(\chi '_{l}(G)=\Delta \) and \(\chi ''_{l}(G)=\Delta +1\) for a planar graph G with maximum degree \(\Delta \ge 8\) and without chordal 6-cycles, where the list edge chromatic number \(\chi '_{l}(G)\) of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number \(\chi ''_{l}(G)\) of G is the smallest integer k such that G is total-k-choosable.

Keywords

List coloring Choosability Planar graph Chordal 

Mathematics Subject Classification

05C15 

References

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  2. 2.Department of MathematicsOcean University of ChinaQingdaoChina
  3. 3.School of ManagementShanghai UniversityShanghaiChina
  4. 4.Department of Computer Science and TechnologyHarbin Institute of Technology Shenzhen Graduate SchoolShenzhenChina
  5. 5.School of MathematicsShandong UniversityJinanChina

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