Unavoidable Induced Subgraphs of Large 2-Connected Graphs

TL;DR

This paper extends classical Ramsey theory results to large 2-connected graphs, showing they necessarily contain certain complex induced subgraphs such as complete graphs, subdivisions, or ladder-like structures.

Contribution

It establishes a new induced subgraph theorem for large 2-connected graphs, identifying specific unavoidable structures analogous to known results for general and connected graphs.

Findings

Large 2-connected graphs contain specific induced subgraphs like $K_n$, subdivisions of $K_{2,n}$, or ladder-like structures.

The result generalizes classical Ramsey-type theorems to the class of 2-connected graphs.

The proof characterizes the structure of large 2-connected graphs and their unavoidable induced subgraphs.

Abstract

Ramsey proved that for every positive integer $n$, every sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive integer $n$, every sufficiently large connected graph contains an induced $K_n$, $K_{1,n}$, or $P_n$. In this paper, we establish an analogue for 2-connected graphs. In particular, we prove that for every integer exceeding two, every sufficiently large 2-connected graph contains one of the following as an induced subgraph: $K_n$, a subdivision of $K_{2,n}$, a subdivision of $K_{2,n}$ with an edge between the two vertices of degree $n$, and a well-defined structure similar to a ladder.