Inequalities for the First-Fit chromatic number

Z. Füredi, A. Gyárfás, Gábor N. Sárközy, Stanley Selkow

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

The First-Fit (or Grundy) chromatic number of G, written as ΧFF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well-known Nordhaus-Gaddum inequality states that the sum of the ordinary chromatic numbers of an n-vertex graph and its complement is at most n + 1. Zaker suggested finding the analogous inequality for the First-Fit chromatic number. We show for n ≥ 10 that ⌊(5n + 2)/4⌋ is an upper bound, and this is sharp. We extend the problem for multicolorings as well and prove asymptotic results for infinitely many cases, We also show that the smallest order of C4-free bipartite graphs with ΧFF(G) = k is asymptotically 2k2 (the upper bound answers a problem of Zaker [9]).

Original languageEnglish
Pages (from-to)75-88
Number of pages14
JournalJournal of Graph Theory
Volume59
Issue number1
DOIs
Publication statusPublished - Sep 2008

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Chromatic number
Upper bound
Vertex of a graph
Independent Set
Bipartite Graph
Complement
Partition
Graph in graph theory

Keywords

  • Chromatic number
  • Difference set
  • Graphs
  • Greedy algorithm
  • Projective planes
  • Quadrilateral-free

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Inequalities for the First-Fit chromatic number. / Füredi, Z.; Gyárfás, A.; Sárközy, Gábor N.; Selkow, Stanley.

In: Journal of Graph Theory, Vol. 59, No. 1, 09.2008, p. 75-88.

Research output: Contribution to journalArticle

Füredi, Z. ; Gyárfás, A. ; Sárközy, Gábor N. ; Selkow, Stanley. / Inequalities for the First-Fit chromatic number. In: Journal of Graph Theory. 2008 ; Vol. 59, No. 1. pp. 75-88.
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