Removable paths and cycles with parity constraints

Abstract

We consider the following problem.

For every positive integer k there is a smallest integer $f(k)$ such that for any two vertices s and t in a non-bipartite $f(k)$-connected graph G, there is an s–t path P in G with specified parity such that $G-V(P)$ is k-connected.

This conjecture is a variant of the well-known conjecture of Lovász with the parity condition. Indeed, this conjecture is strictly stronger. Lovász' conjecture is wide open for $k⩾3$.

In this paper, we show that $f(1)=5$ and $6⩽f(2)⩽8$.

We also consider a conjecture of Thomassen which says that there exists a function $f(k)$ such that every $f(k)$-connected graph with an odd cycle contains an odd cycle C such that $G-V(C)$ is k-connected. We show the following strengthening of Thomassen's conjecture for the case $k=2$. Namely; let G be a 5-connected graph and s be a vertex in G such that $G-s$ is not bipartite. Then there is an odd cycle C avoiding s such that $G-V(C)$ is 2-connected.