The Global Structure of a Typical Graph Without $H$ as an Induced Subgraph when $H$ is a Cycle
Abstract
One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two. We show that almost every graph which does not contain a cycle of length six as an induced subgraph has such a certificate. We obtain similar characterizations of the structure of almost all graphs which contain no induced cycle of length for all even exceeding six. (Similar results were obtained for by Erdos, Kleitman, and Rothschild in 1976, for by Promel and Steger in 1991 and for odd exceeding 5 by Balogh and Butterfield in 2009.) We prove that a simiiar theorem for all holds up to the deletion of a set of vertices and ask for which the characterization holds fully.
Cite
@article{arxiv.2506.03544,
title = {The Global Structure of a Typical Graph Without $H$ as an Induced Subgraph when $H$ is a Cycle},
author = {Bruce Reed},
journal= {arXiv preprint arXiv:2506.03544},
year = {2025}
}