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Choice Numbers of Graphs: a Probabilistic Approach

Published online by Cambridge University Press:  12 September 2008

Noga Alon
Noga Alon
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and Bellcore, Morristown, NJ 07960, USA
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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).

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 2 , June 1992 , pp. 107 - 114
Copyright
Copyright © Cambridge University Press 1992

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References

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