Sparse graphs usually have exponentially many optimal colorings

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Abstract

A proper coloring of a graph G = (V,E) is called optimal if the number of colors used is the minimal possible; i.e., it coincides with the chromatic number of G. We investigate the typical behavior of the number of distinct optimal colorings of a random graph G(n, p), for various values of the edge probability p = p(n). Our main result shows that for every constant 1/3 < a < 2, most of the graphs in the probability space G(n, p) with p = n -a have exponentially many optimal colorings.

Original languageEnglish
Pages (from-to)1-8
Number of pages8
JournalElectronic Journal of Combinatorics
Volume9
Issue number1 R
DOIs
StatePublished - 2002

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