Random graphs and Lindstrom quantifiers for natural graph properties

TL;DR

This paper investigates the expressibility of graph properties within logical languages under zero-one laws, showing that some properties like connectivity obey the law, while others like Hamiltonicity do not, due to their interpretability of arithmetic.

Contribution

It demonstrates the existence of logical languages that can express certain graph properties while still satisfying the zero-one law, and others that cannot, clarifying the limits of expressibility.

Findings

Languages expressing connectivity obey the zero-one law.

Languages expressing Hamiltonicity interpret arithmetic, violating the zero-one law.

The results answer a question posed by Blass and Harary.

Abstract

We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P, is there a language of graphs able to express P while obeying the zero-one law? Our results show that on the one hand there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we show that in any (semiregular) language strong enough to express Hamiltonicity one can interpret arithmetic and thus the zero-one law fails miserably. This answers a question of Blass and Harary.