### 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 language | English |
---|---|

Pages (from-to) | 621-632 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 338 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 6 2015 |

### Fingerprint

### 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

*Discrete Mathematics*,

*338*(4), 621-632. https://doi.org/10.1016/j.disc.2014.12.002

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

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 338, no. 4, pp. 621-632. https://doi.org/10.1016/j.disc.2014.12.002

}

TY - JOUR

T1 - Minimum order of graphs with given coloring parameters

AU - Bacsó, Gábor

AU - Borowiecki, Piotr

AU - Hujter, Mihály

AU - Tuza, Z.

PY - 2015/4/6

Y1 - 2015/4/6

N2 - 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.

AB - 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.

KW - Achromatic number

KW - Bipartite graph

KW - Extremal graph

KW - Graph coloring

KW - Greedy algorithm

KW - Grundy number

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UR - http://www.scopus.com/inward/citedby.url?scp=84919907632&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2014.12.002

DO - 10.1016/j.disc.2014.12.002

M3 - Article

VL - 338

SP - 621

EP - 632

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -