A note on Lovász removable path conjecture

Abstract

Lovasz [8] conjectured that for any natural number k, there exists a least natural number f(k) such that, for any two vertices s, t in any f(k)-connected graph G, there exists an s-t path P such that G−V (P ) is k-connected. This conjecture is proved only for k ≤ 2. Here, we strengthen the result for k = 2 as follows: for any integers l > 0 and m ≥ 0, there exists a function f(l, m) such that, for any distinct vertices s, t, v1, ..., vm in any f(l, m)-connected graph G, there exist l internally vertex disjoint s-t paths P1, ..., Pl such that for any subset I ⊂ {1, ..., l}, G− ∪i∈IV (Pi) is 2-connected and {v1, v2, ..., vm} ⊂ V (G)− ∪1≤i≤lV (Pi).