Mader's conjecture for graphs with small connectivity

TL;DR

This paper confirms Mader's conjecture for small connectivity values by characterizing subgraphs containing specific trees while avoiding certain vertices, advancing understanding of tree embeddings in highly connected graphs.

Contribution

It provides a characterization for subgraphs containing specified trees avoiding certain vertices and confirms Mader's conjecture for k ≤ 3.

Findings

Confirmed Mader's conjecture for k ≤ 3

Provided a new characterization for subgraph embeddings

Enhanced understanding of tree embeddings in k-connected graphs

Abstract

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a characterization for a subgraph to contain an embedding of a specified tree avoiding some vertex. As a corollary, we confirm Mader's conjecture for $k\leq3$.