On the Typical Structure of Graphs in a Monotone Property

Svante Janson, Andrew J. Uzzell

Abstract


Given a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$.  In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs.  Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.

Keywords


Graph limits; Monotone properties; Structure of graphs

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