The Global Structure of a Typical Graph Without $H$ as an Induced Subgraph when $H$ is a Cycle

TL;DR

This paper characterizes the typical structure of graphs that exclude certain induced cycles, showing that almost all such graphs have a specific partition structure, extending previous results for various cycle lengths.

Contribution

It provides a new structural characterization for almost all graphs excluding even cycles of length greater than six, generalizing earlier cycle-specific results.

Findings

Almost all graphs without a 6-cycle have a specific partition structure.

Similar characterizations are extended to all even cycles exceeding six.

The paper proposes a conjecture for full characterization for all $H$.

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 $k$ for all even $k$ exceeding six. (Similar results were obtained for $k=3$ by Erdos, Kleitman, and Rothschild in 1976, for $k =4,5$ by Promel and Steger in 1991 and for odd $k$ exceeding 5 by Balogh and Butterfield in 2009.) We prove that a simiiar theorem for all $H$ holds up to the deletion of a set of $o(|V(G)|)$ vertices and ask for which $H$ the characterization holds fully.