Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)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 Lovasz with the parity condition. Indeed, this conjecture is strictly stronger. Lovasz' 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. (C) 2014 Elsevier Inc. All rights reserved.106115133Japan Society for the Promotion of ScienceKayamori FoundationJST, ERATO, Kawarabayashi Large Graph ProjectConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)[Proc. 477692/2012-5]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [Proc. 301310/2005-0, Proc. 472504/2007-0, Proc. 473867/2010-9][Proc. 477692/2012-5]