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A Limit Law of Almost l-partite Graphs

Published online by Cambridge University Press:  12 March 2014

Vera Koponen
Vera Koponen*
Affiliation:
Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden, E-mail: vera@math.uu.se
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Abstract

For integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1, …, n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn (l, d). We prove a labelled first-order limit law, i.e., for every first-order sentence φ, the proportion of graphs in Pn (l, d) that satisfy φ converges as n → ∞. By combining this result with a result of Hundack, Prömel and Steger [12] we also prove that if 1 ≤ s1 ≤ … ≤ sl are integers, then Forb() has a labelled first-order limit law, where Forb() denotes the set of all graphs with vertices 1, …, n, for some n, in which there is no subgraph isomorphic to the complete (l + 1 )-partite graph with parts of sizes 1, s1, …, sl. In the course of doing this we also prove that there exists a first-order formula ξ, depending only on l and d, such that the proportion of ∈ Pn (l, d) with the following property approaches 1 as n → ∞: there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by ξ.

Keywords

Finite model theorylimit lawrandom graphforbidden subgraph
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 911 - 936
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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