# Cycles intersecting edge-cuts of prescribed sizes

Abstract : We prove that every cubic bridgeless graph $G$ contains a $2$-factor which intersects all (minimal) edge-cuts 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 edge-disjoint circuits) that intersects all edge-cuts 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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Communication dans un congrès
Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.303-308, 2005, DMTCS Proceedings
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https://hal.inria.fr/hal-01184454
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• HAL Id : hal-01184454, version 1

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Tomáš Kaiser, Riste Škrekovski. Cycles intersecting edge-cuts of prescribed sizes. Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.303-308, 2005, DMTCS Proceedings. 〈hal-01184454〉

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