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Separation Choosability and Dense Bipartite Induced Subgraphs

Part of: Graph theory

Published online by Cambridge University Press:  26 February 2019

Louis Esperet ,
Ross J. Kang  and
Stéphan Thomassé
Louis Esperet
Affiliation:
Université Grenoble Alpes, CNRS, G-SCOP, 46 Avenue Félix Viallet, 38000 Grenoble, France
Ross J. Kang*
Affiliation:
Radboud University Nijmegen, PO box 9010, 6500 GL Nijmegen, Netherlands
Stéphan Thomassé
Affiliation:
Laboratoire d’Informatique du Parallélisme, École Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon, France
*
*Corresponding author. Email: r.kang@math.ru.nl
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Abstract

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We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 5 , September 2019 , pp. 720 - 732
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge University Press 2019

Footnotes

†

This research was partly supported by a Van Gogh grant, reference 35513NM.

‡

This author is partially supported by ANR Projects STINT (anr-13-bs02-0007) and GATO (anr-16-ce40-0009-01), and LabEx PERSYVAL-Lab (anr-11-labx-0025).

§

This author is partially supported by a Vidi grant (639.032.614) of the Netherlands Organisation for Scientific Research (NWO).

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