# Choice Numbers of Graphs

## A Probabilistic Approach

Research output: Contribution to journalArticle

34 Citations (Scopus)

### Abstract

The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

Original language English (US) 107-114 8 Combinatorics, Probability and Computing 1 2 https://doi.org/10.1017/S0963548300000122 Published - Jan 1 1992

### Fingerprint

Probabilistic Approach
Color
Coloring
Graph in graph theory
Vertex of a graph
Probabilistic Methods
Colouring
Assign
Assignment
Complement
Imply
Integer

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Applied Mathematics
• Statistics and Probability
• Computational Theory and Mathematics

### Cite this

title = "Choice Numbers of Graphs: A Probabilistic Approach",
abstract = "The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).",
author = "Alon, {Noga Mordechai}",
year = "1992",
month = "1",
day = "1",
doi = "https://doi.org/10.1017/S0963548300000122",
language = "English (US)",
volume = "1",
pages = "107--114",
journal = "Combinatorics Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "2",

}

In: Combinatorics, Probability and Computing, Vol. 1, No. 2, 01.01.1992, p. 107-114.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Choice Numbers of Graphs

T2 - A Probabilistic Approach

AU - Alon, Noga Mordechai

PY - 1992/1/1

Y1 - 1992/1/1

N2 - The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

AB - The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). By applying probabilistic methods, it is shown that there are two positive constants c1 and c2 such that for all m ≥ 2 and r ≥ 2 the choice number of the complete r-partite graph with m vertices in each vertex class is between c1r log m and c2r log m. This supplies the solutions of two problems of Erdős, Rubin and Taylor, as it implies that the choice number of almost all the graphs on n vertices is o(n) and that there is an n vertex graph G such that the sum of the choice number of G with that of its complement is at most O(n1/2(log n)1/2).

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U2 - https://doi.org/10.1017/S0963548300000122

DO - https://doi.org/10.1017/S0963548300000122

M3 - Article

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SP - 107

EP - 114

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -