The Structure of Hereditary Properties and Colourings of Random Graphs
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in the probability space \(\)? Second, does there exist a constant \(\) such that the \(\)-chromatic number of the random graph \(\) is almost surely \(\)? The second question was posed by Scheinerman (SIAM J. Discrete Math.5 (1992) 74–80).
The two questions are closely related and, in the case p=1/2, have already been answered. Prömel and Steger (Contemporary Mathematics147, Amer. Math. Soc., Providence, 1993, pp. 167-178), Alekseev (Discrete Math. Appl.3 (1993) 191-199) and the authors ( Algorithms and Combinatorics14 Springer-Verlag (1997) 70–78) provided an approximation which was used by the authors (Random Structures and Algorithms6 (1995) 353–356) to answer the \(\)-chromatic question for p=1/2. However, the approximating properties that work well for p=1/2 fail completely for \(\).
In this paper we describe a class of properties that do approximate \(\) in \(\), in the following sense: for any desired accuracy of approximation, there is a property in our class that approximates \(\) to this level of accuracy. As may be expected, our class includes the simple properties used in the case p=1/2.
The main difficulty in answering the second of our two questions, that concerning the \(\)-chromatic number of \(\), is that the number of small \(\)-graphs in \(\) has, in general, large variance. The variance is smaller if we replace \(\) by a simple approximation, but it is still not small enough. We overcome this by considering instead a very rigid non-hereditary subproperty \(\) of the approximating property; the variance of the number of small \(\)-graphs is small enough for our purpose, and the structure of \(\) is sufficiently restricted to enable us to show this by a fine analysis.
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