Connectivity keeping caterpillars and spiders in bipartite graphs with connectivity at most three

TL;DR

This paper confirms a conjecture about embedding certain trees in highly connected bipartite graphs with minimum degree constraints, specifically for caterpillars and spiders when the connectivity is at most three.

Contribution

It proves the conjecture for caterpillars with k=3 and spiders with k ≤ 3, advancing understanding of tree embeddings in bipartite graphs under connectivity constraints.

Findings

Confirmed the conjecture for caterpillars at k=3.

Confirmed the conjecture for spiders at k≤3.

Established conditions for subtree embeddings in bipartite graphs.

Abstract

A conjecture of Luo, Tian and Wu (2022) says that for every positive integer $k$ and every finite tree $T$ with bipartition $X$ and $Y$ (denote $t = \max\{|X|,|Y |\})$, every $k$-connected bipartite graph $G$ with $\delta(G) \geq k + t$ contains a subtree $T' \cong T$ such that $\kappa(G-V (T')) \geq k$. In this paper, we confirm this conjecture for caterpillars when $k=3$ and spiders when $k\leq 3$.