### Abstract

Given a graph G = (V, E) and sets L (v) of allowed colors for each v ∈ V, a list coloring of G is an assignment of colors φ (v) to the vertices, such that φ (v) ∈ L (v) for all v ∈ V and φ (u) ≠ φ (v) for all u v ∈ E. The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L (v) | ≥ k holds for every vertex v. This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L (v). (On complete graphs, this condition is equivalent to the well-known one in Hall's Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n - 3. This estimate is tight for all n. Tightness is deduced from the upper bound that every graph of order n has Hall number at most n - 2. We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K_{2} ∪ (n - 2) K_{1}, P_{4} ∪ (n - 4) K_{1}, and C_{5} ∪ (n - 5) K_{1}. Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161-182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227-245] as a function of maximum degree.

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

Pages (from-to) | 461-470 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 310 |

Issue number | 3 |

DOIs | |

Publication status | Published - Feb 6 2010 |

### Fingerprint

### Keywords

- Graph
- Hall condition
- Hall number
- List coloring
- Vertex coloring

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Hall number for list colorings of graphs : Extremal results.** / Tuza, Zsolt.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 310, no. 3, pp. 461-470. https://doi.org/10.1016/j.disc.2009.03.025

}

TY - JOUR

T1 - Hall number for list colorings of graphs

T2 - Extremal results

AU - Tuza, Zsolt

PY - 2010/2/6

Y1 - 2010/2/6

N2 - Given a graph G = (V, E) and sets L (v) of allowed colors for each v ∈ V, a list coloring of G is an assignment of colors φ (v) to the vertices, such that φ (v) ∈ L (v) for all v ∈ V and φ (u) ≠ φ (v) for all u v ∈ E. The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L (v) | ≥ k holds for every vertex v. This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L (v). (On complete graphs, this condition is equivalent to the well-known one in Hall's Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n - 3. This estimate is tight for all n. Tightness is deduced from the upper bound that every graph of order n has Hall number at most n - 2. We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K2 ∪ (n - 2) K1, P4 ∪ (n - 4) K1, and C5 ∪ (n - 5) K1. Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161-182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227-245] as a function of maximum degree.

AB - Given a graph G = (V, E) and sets L (v) of allowed colors for each v ∈ V, a list coloring of G is an assignment of colors φ (v) to the vertices, such that φ (v) ∈ L (v) for all v ∈ V and φ (u) ≠ φ (v) for all u v ∈ E. The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L (v) | ≥ k holds for every vertex v. This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L (v). (On complete graphs, this condition is equivalent to the well-known one in Hall's Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n - 3. This estimate is tight for all n. Tightness is deduced from the upper bound that every graph of order n has Hall number at most n - 2. We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K2 ∪ (n - 2) K1, P4 ∪ (n - 4) K1, and C5 ∪ (n - 5) K1. Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161-182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227-245] as a function of maximum degree.

KW - Graph

KW - Hall condition

KW - Hall number

KW - List coloring

KW - Vertex coloring

UR - http://www.scopus.com/inward/record.url?scp=70450267698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70450267698&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2009.03.025

DO - 10.1016/j.disc.2009.03.025

M3 - Article

AN - SCOPUS:70450267698

VL - 310

SP - 461

EP - 470

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

ER -