Removable paths and cycles with parity constraints

We consider the following problem. For every positive integer k   there is a smallest integer f(k)f(k) such that for any two vertices s and t   in a non-bipartite f(k)f(k)-connected graph G, there is an s–t path P in G   with specified parity such that G−V(P)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⩾3k⩾3.In this paper, we show that f(1)=5f(1)=5 and 6⩽f(2)⩽86⩽f(2)⩽8.We also consider a conjecture of Thomassen which says that there exists a function f(k)f(k) such that every f(k)f(k)-connected graph with an odd cycle contains an odd cycle C   such that G−V(C)G−V(C) is k  -connected. We show the following strengthening of Thomassen's conjecture for the case k=2k=2. Namely; let G be a 5-connected graph and s be a vertex in G   such that G−sG−s is not bipartite. Then there is an odd cycle C avoiding s   such that G−V(C)G−V(C) is 2-connected.