Connectivity keeping paths containing prescribed vertices in highly connected triangle-free graphs

Abstract

Let $m,k$ be integers with $m\ge 1,k\ge 2$. For a k-connected graph G, a subgraph H of G is k-removable if $G-V\left(H\right)$ is still a k-connected graph. A graph is triangle-free if it contains no triangle as a subgraph.

In this paper, we prove that if G is a k-connected triangle-free graph with minimum degree at least $k+(m-1)/2$, then for any vertex $v\in V\left(G\right)$, there exists a path P on m vertices starting from v such that $G-V\left(P\right)$ is a $(k-1)$-connected graph. This result is obtained by showing a stronger statement concerning the existence of k-removable paths in k-connected triangle-free graphs. We also prove that if G is a k-connected triangle-free graph with minimum degree at least $k+1$, then G contains a k-removable edge. Our results confirm a conjecture due to Luo et al. concerning the existence of a k-removable path on m vertices in a k-connected bipartite graph for all odd m together with the case $m=2$.