On choosability with separation of planar graphs with lists of different sizes

Henry Kierstead, Bernard Lidický

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Abstract A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. It is known that planar graphs are (4, 1)-choosable but it is not known if planar graphs are (3, 1)-choosable. We strengthen the result that planar graphs are (4, 1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4. Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors.

Original languageEnglish (US)
Article number10037
Pages (from-to)1779-1783
Number of pages5
JournalDiscrete Mathematics
Volume338
Issue number10
DOIs
StatePublished - Sep 30 2015

Keywords

  • Choosability with separation
  • Graph coloring
  • List coloring
  • Planar graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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