Domination and dominator colorings in planar graphs with small diameter

Wayne Goddard, Michael A. Henning

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

A dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class, where a vertex dominates itself and all vertices adjacent to it. The dominator chromatic number of G is the minimum number of colors among all dominator coloring of G. A total dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class other than its own. The total dominator chromatic number of G is the minimum number of colors among all total dominator coloring of G. In this paper, we present bounds on the dominator chromatic number and total dominator chromatic number of a planar graphs with small diameter. In particular we show the dominator chromatic number of a planar graph of diameter 2 is at most 5. We also present results for the special cases of outerplanar graphs and bipartite planar graphs.

Original languageEnglish
Pages (from-to)80-92
Number of pages13
JournalDiscrete Applied Mathematics
Volume313
DOIs
Publication statusPublished - 31 May 2022

Keywords

  • Domination
  • Dominator coloring
  • Graph colorings
  • Total dominator coloring

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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