Connectivity keeping paths in $k$-connected bipartite graphs

TL;DR

This paper extends a connectivity preservation result from general graphs to bipartite graphs, showing that certain minimum degree conditions guarantee the existence of paths that leave the graph still k-connected.

Contribution

It establishes a new minimum degree condition for bipartite graphs to contain paths that preserve k-connectivity after removal.

Findings

Proves a minimum degree condition of at least k+m for bipartite graphs.

Shows the existence of a path of order m maintaining k-connectivity.

Extends Mader's 2010 result to bipartite graphs.

Abstract

In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every $k$-connected bipartite graph $G$ with minimum degree at least $k+m$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected.