Proof a conjecture on connectivity keeping odd paths in k-connected bipartite graphs

TL;DR

This paper proves a conjecture about the existence of certain paths in bipartite graphs that preserve connectivity, specifically confirming it for odd-length paths and nearly for even-length paths.

Contribution

The paper confirms the conjecture for paths of odd order and provides a near-complete result for even order paths in bipartite graphs.

Findings

Confirmed the conjecture for odd-length paths in bipartite graphs.

Established a minimum degree condition for paths that preserve k-connectivity.

Extended understanding of connectivity preservation in bipartite graphs.

Abstract

Luo, Tian and Wu (2022) conjectured that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+t$, where $t=$max$\{|X|,|Y|\}$, contains a tree $T'\cong T$ such that $G-V(T')$ is still $k$-connected. Note that $t=\lceil\frac{m}{2}\rceil$ when the tree $T$ is the path with order $m$. In this paper, we proved that every $k$-connected bipartite graph $G$ with minimum degree at least $k+ \lceil\frac{m+1}{2}\rceil$ contains a path $P$ of order $m$ such that $G-V(P)$ remains $k$-connected. This shows that the conjecture is true for paths with odd order. And for paths with even order, the minimum degree bound in this paper is the bound in the conjecture plus one.