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List packing number of bounded degree graphs

Part of: Graph theory

Published online by Cambridge University Press:  18 September 2024

Stijn Cambie Open the ORCID record for Stijn Cambie [Opens in a new window] ,
Wouter Cames van Batenburg ,
Ewan Davies  and
Ross J. Kang
Stijn Cambie*
Affiliation:
Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea Department of Computer Science, KU Leuven Campus Kulak-Kortrijk, 8500 Kortrijk, Belgium
Wouter Cames van Batenburg
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Netherlands
Ewan Davies
Affiliation:
Department of Computer Science, Colorado State University, Fort Collins, USA
Ross J. Kang
Affiliation:
Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Netherlands
*
Corresponding author: Stijn Cambie; Email: stijn.cambie@hotmail.com
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Abstract

We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $\Delta$ has list packing number at most $\Delta +1$. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.

Keywords

Packing of list colouringslist packing numberlist colouringcorrespondence colouringmaximum degreetransversals

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C07: Vertex degrees 05C35: Extremal problems 05C69: Dominating sets, independent sets, cliques 05C70: Factorization, matching, partitioning, covering and packing

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 33 , Issue 6 , November 2024 , pp. 807 - 828
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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