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arXiv · 2602.24064

Determining Factorial Speed Fast

Zhidan Feng·Henning Fernau·Pamela Fleischmann·Philipp Kindermann·Silas Cato Sacher
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Abstract

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{\Theta(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{\Theta(n)}$.

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Zhidan Feng, Henning Fernau, Pamela Fleischmann, Philipp Kindermann, Silas Cato Sacher. 2026-02-27. Determining Factorial Speed Fast. https://arxiv.org/abs/2602.24064

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