The structure and the number of $P_7$-free bipartite graphs

TL;DR

This paper proves that the number of labeled $P_7$-free bipartite graphs grows exponentially with respect to the number of vertices, resolving an open problem and introducing a new graph decomposition method.

Contribution

It establishes the growth rate of $P_7$-free bipartite graphs and introduces a novel decomposition scheme for bipartite graphs.

Findings

Number of labeled $P_7$-free bipartite graphs grows as $n^{ heta(n)}$

Resolves an open problem in graph theory

Introduces a new bipartite graph decomposition scheme

Abstract

We show that the number of labelled $P_7$-free bipartite graphs with $n$ vertices grows as $n^{\Theta(n)}$. This resolves an open problem posed by Allen [P. Allen, Forbidden induced bipartite graphs. J. Graph Theory 60 (2009), no. 3, 219--241.], and completes the description of speeds of monogenic classes of bipartite graphs. Our solution is based on a new decomposition scheme of bipartite graphs, which is of independent interest.