### 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 C_{4}-free bipartite graphs with _{ΧFF}(G) = k is asymptotically 2k^{2} (the upper bound answers a problem of Zaker [9]).

Original language | English |
---|---|

Pages (from-to) | 75-88 |

Number of pages | 14 |

Journal | Journal of Graph Theory |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 2008 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*59*(1), 75-88. https://doi.org/10.1002/jgt.20327

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

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 59, no. 1, pp. 75-88. https://doi.org/10.1002/jgt.20327

}

TY - JOUR

T1 - Inequalities for the First-Fit chromatic number

AU - Füredi, Z.

AU - Gyárfás, A.

AU - Sárközy, Gábor N.

AU - Selkow, Stanley

PY - 2008/9

Y1 - 2008/9

N2 - 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]).

AB - 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]).

KW - Chromatic number

KW - Difference set

KW - Graphs

KW - Greedy algorithm

KW - Projective planes

KW - Quadrilateral-free

UR - http://www.scopus.com/inward/record.url?scp=51049087097&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51049087097&partnerID=8YFLogxK

U2 - 10.1002/jgt.20327

DO - 10.1002/jgt.20327

M3 - Article

AN - SCOPUS:51049087097

VL - 59

SP - 75

EP - 88

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -