2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 303308
author:  Tomáš Kaiser and Riste Škrekovski 

title:  Cycles intersecting edgecuts of prescribed sizes 
keywords:  graph, cycle, edgecut, covering cycle, coverable set 
abstract: 
We prove that every cubic bridgeless graph
G
contains a
2
factor which intersects all (minimal) edgecuts of
size
3
or
4
. This generalizes an earlier result of the authors,
namely that such a
2
factor exists provided that
G
is planar. As a further extension, we show that every
graph contains a cycle (a union of edgedisjoint circuits)
that intersects all edgecuts of size
3
or
4
. Motivated by this result, we introduce the concept
of a coverable set of integers and discuss a number of
questions, some of which are related to classical problems
of graph theory such as Tutte's
4
flow conjecture or the Dominating circuit
conjecture.

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reference:  Tomáš Kaiser and Riste Škrekovski (2005), Cycles intersecting edgecuts of prescribed sizes, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 303308 
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