Abstract
We consider the following problem. 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.
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