Non-separating subgraphs after deleting many disjoint paths

Abstract

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 $P_1,\dots ,P_k$ with endpoints s and t such that $G-{\bigcup }_{i=1}^{k}V(P_i)$ 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].