Non-separating subgraphs after deleting many disjoint paths

Motivated by the well-known conjecture by Lovász (1975) [6] on the connectivity after the path removal, we study the following problem:There exists a function f=f(k,l) such that the following holds. For every f(k,l)-connected graph G and two distinct vertices s and t in G, there are k internally disjoint paths P1,…,Pk with endpoints s and t such that is l-connected.When k=1, this problem corresponds to Lovász conjecture, and it is open for all the cases l⩾3.We show that f(k,1)=2k+1 and f(k,2)⩽3k+2. The connectivity “2k+1” for f(k,1) is best possible. Thus our result generalizes the result by Tutte (1963) [8], for the case k=1 and l=1 (the first settled case of Lovász conjecture), and the result by Chen, Gould and Yu (2003) [1], , Kriesell (2001) [4], , Kawarabayashi, Lee, and Yu (2005) [2], independently, for the case k=1 and l=2 (the second settled case of Lovász conjecture).When l=1, our result also improves the connectivity bound “22k+2” given by Chen, Gould and Yu (2003) [1].