Matrix-free proof of a regularity characterization
Summary: The central concept in Szemerédi's powerful regularity lemma is the so-called \(\varepsilon\)-regular pair. A useful statement of \textit{N. Alon} et al. [J. Algorithms 16, 80-109 (1994; Zbl 0794.05119)] essentially equates the notion of an \(\varepsilon\)-regular pair with degree uniformity of vertices and pairs of vertices. The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix argument of Alon et al. We show the \(\varepsilon\)-regular characterization follows from an application of Szemerédi's regularity lemma itself.
scientific article; zbMATH DE number 2038337