Unavoidable induced subgraphs of large and infinite $2$-edge-connected graphs

TL;DR

This work establishes 2-edge-connected analogues of Ramsey-type results for unavoidable large induced subgraphs in both finite and infinite graphs. It introduces and employs new structural families—such as $r$-flowers, clean and pinched ladders, and chains of super-clean pinched ladders—along with a super-cleaning procedure to derive induced subgraphs that must appear in large graphs. The authors derive comprehensive consequences for subgraphs, topological minors, minors, induced topological minors, induced minors, and Eulerian subgraphs, and extend these results to multigraphs. The results unify Ramsey-type unavoidable structures under $2$-edge-connectivity and pave the way for further exploration in higher connectivity and alternative graph orderings.

Abstract

In 1930, Ramsey proved that every large graph contains either a large clique or a large edgeless graph as an induced subgraph. It is well known that every large connected graph contains a long path, a large clique, or a large star as an induced subgraph. Recently Allred, Ding, and Oporowski presented the unavoidable large induced subgraphs for large $2$-connected graphs and for infinite $2$-connected graphs. In this paper we establish the $2$-edge-connected analogues of these results. As consequences we obtain results on unavoidable large subgraphs, topological minors, minors, induced topological minors, induced minors, and Eulerian subgraphs in large and infinite $2$-edge-connected graphs. When appropriate we extend our results to multigraphs.

Figures (6)

• Figure 1.1: Structure of $\Theta_{\infty}$
• Figure 1.2: Two types of fan-like structure
• Figure 1.3: Ladder-like structures
• Figure 1.4: Graphs from \ref{['thm:2econ']}
• Figure 1.5: A chain of $5$ super-clean pinched ladders

Theorems & Definitions (39)

• Theorem 1.1: König konig
• Theorem 1.2: Ramsey ramsey
• Theorem 1.3
• Theorem 1.4: Ramsey ramsey
• Theorem 1.5
• Theorem 1.6: Unavoidableminors3connbinmatroids or diestel
• Theorem 1.7: unavoidableinducedsubgraphs
• Theorem 1.8: unavoidableinfinducedsubgraphs
• Theorem 1.9
• Theorem 1.10