### 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 c_{1} and c_{2} 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 c_{1}r log m and c_{2}r 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(n^{1/2}(log n)^{1/2}).

Original language | English (US) |
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Pages (from-to) | 107-114 |

Number of pages | 8 |

Journal | Combinatorics, Probability and Computing |

Volume | 1 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1992 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Combinatorics, Probability and Computing*, vol. 1, no. 2, pp. 107-114. https://doi.org/10.1017/S0963548300000122

**Choice Numbers of Graphs : A Probabilistic Approach.** / Alon, Noga Mordechai.

Research output: Contribution to journal › Article

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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UR - http://www.scopus.com/inward/citedby.url?scp=84971769194&partnerID=8YFLogxK

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

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

M3 - Article

VL - 1

SP - 107

EP - 114

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -