Minimum order of graphs with given coloring parameters

Gábor Bacsó, Piotr Borowiecki, Mihály Hujter, Z. Tuza

Research output: Contribution to journalArticle

Abstract

A complete k-coloring of a graph G=(V,E) is an assignment φ:V→{1,⋯,k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number χ(G) and achromatic number ψ(G), respectively), and the Grundy number Γ(G) defined as the largest k admitting a complete coloring φ with exactly k colors such that every vertex v→V of color φ(v) has a neighbor of color i for all 1≤lt(v). The inequality chain χ(G)≤Γ(G)≤(G) obviously holds for all graphs G. A triple (f,g,h) of positive integers at least 2 is called realizable if there exists a connected graph G with χ(G)=f, Γ(G)=g, and ψ(G)=h. In Chartrand et al. (2010), the list of realizable triples has been found. In this paper we determine the minimum number of vertices in a connected graph with chromatic number f, Grundy number g, and achromatic number h, for all realizable triples (f,g,h) of integers. Furthermore, for f=g=3 we describe the (two) extremal graphs for each h≥6. For h→{4,5}, there are more extremal graphs, their description is given as well.

Original languageEnglish
Pages (from-to)621-632
Number of pages12
JournalDiscrete Mathematics
Volume338
Issue number4
DOIs
Publication statusPublished - Apr 6 2015

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Coloring
Colouring
Color
Graph in graph theory
Achromatic number
Extremal Graphs
Chromatic number
Connected graph
Graph Invariants
Integer
Union
Assignment
Adjacent
Vertex of a graph

Keywords

  • Achromatic number
  • Bipartite graph
  • Extremal graph
  • Graph coloring
  • Greedy algorithm
  • Grundy number

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Minimum order of graphs with given coloring parameters. / Bacsó, Gábor; Borowiecki, Piotr; Hujter, Mihály; Tuza, Z.

In: Discrete Mathematics, Vol. 338, No. 4, 06.04.2015, p. 621-632.

Research output: Contribution to journalArticle

Bacsó, Gábor ; Borowiecki, Piotr ; Hujter, Mihály ; Tuza, Z. / Minimum order of graphs with given coloring parameters. In: Discrete Mathematics. 2015 ; Vol. 338, No. 4. pp. 621-632.
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