An Improved Interpolation Theorem and Disproofs of Two Conjectures on 2-Connected Subgraphs

TL;DR

This paper improves a known result on the existence of 2-connected subgraphs in graphs with certain minimum degree conditions, disproves two conjectures on 2-connected subgraphs using counterexamples and combinatorial designs, and proposes a new conjecture.

Contribution

It advances the understanding of 2-connected subgraph existence, disproves two conjectures with explicit counterexamples, and introduces a new conjecture relating minimum degree to subgraph connectivity.

Findings

Improved minimum degree condition guarantees 2-connected subgraphs of all sizes.

Constructed counterexamples disproving two existing conjectures.

Proposed a new conjecture linking minimum degree and subgraph connectivity.

Abstract

We prove that any \(2\)-connected graph \(G\) on \(n\) vertices with minimum degree \(\delta(G) \ge \frac{n}{4}+2\) contains a \(2\)-connected subgraph of order \(k\) for every integer \(k\) with \(4 \le k \le n\). This improves a previous result of Yin and Wu. In \cite{YinWu-DAM-2026}, Yin and Wu proposed two conjectures. The first states that for any \(2\)-connected graph \(G\) of order \(n\) and size \(m\), there exists a \(2\)-connected subgraph of order \(k\) for each \(k \in \{4, \dots, n\}\) whenever \(m \ge \frac{1}{2} n^{3/2}\). The second conjecture asserts that the same conclusion holds under the alternative condition \(\delta(G) \ge \sqrt{n}\). In this paper, we construct counterexamples that completely disprove the first conjecture. Furthermore, using the existence of \((v, k, 2)\)-Symmetric Balanced Incomplete Block designs (i.e., SBIBDs), we disprove the second conjecture for all \(n \in \{8, 14, 22, 32, 74, 112, 158\}\). Finally, we propose a conjecture of our own: for any \(2\)-connected graph \(G\) on \(n\) vertices with \(\delta(G) \ge \frac{n}{k}\), where \(k \ge 3\) and \(n\) is sufficiently large, \(G\) contains a \(2\)-connected subgraph of every order from \(4\) to \(n\).