Home » Subject » Graph Theory
Importance: Medium ✭✭
Author(s): Lovasz, Laszlo
Subject: Graph Theory
Keywords:
Recomm. for undergrads: no
Posted by: fhavet
on: March 4th, 2013
Conjecture   There is an integer-valued function $ f(k) $ such that if $ G $ is any $ f(k) $-connected graph and $ x $ and $ y $ are any two vertices of $ G $, then there exists an induced path $ P $ with ends $ x $ and $ y $ such that $ G-V(P) $ is $ k $-connected.

It follows from a theorem of Tutte that any 3-connected graph contains a non-separating path connecting any two vertices, and consequently, $ f(1)=3 $. When $ k=2 $, it was independently shown by Chen, Gould, and Yu [CGY] and Kriesell [K] that $ f(2) = 5 $.

Anwering a conjecture of Kriesell, Kawarabayashi et al. [KLRW] proved the following weaker statement, in which one only removes the edges of the path.

Theorem   There exists a function $ f(k) $ such that for every $ f(k) $-connected graph $ G $ and any two vertices $ x $ and $ y $ of $ G $, there exists an induced path $ P $ with ends $ x $ and $ y $ such that $ G\setminus E(P) $ is $ k $-connected.

Bibliography

[CGY] G. Chen, R. Gould, X. Yu, Graph connectivity after path removal, Combinatorica 23 (2003) 185--203.

[KLRW] K. Kawarabayashi, O. Lee, B. Reed, and P. Wollan, A weaker version of Lovasz's path removal conjecture, Journal of Combinatorial Theory, Series B 98 (2008) 972--979.

[K] M. Kriesell, Induced paths in 5-connected graphs, J. of Graph Theory, 36 (2001), 52--58.

*[T] C. Thomassen, Graph decompositions with applications to subdivisions and path systems modulo k, J. of Graph Theory, 7 (1983), 261--271.


* indicates original appearance(s) of problem.
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