par Cames Van Batenburg, Wouter 
;Joret, Gwenaël ;Ulmer, Arthur
Référence SIAM journal on discrete mathematics, 34, 3, page (1609-1619)
Publication Publié, 2020-12-01
;Joret, Gwenaël ;Ulmer, ArthurRéférence SIAM journal on discrete mathematics, 34, 3, page (1609-1619)
Publication Publié, 2020-12-01
Article révisé par les pairs
| Résumé : | A classic theorem of Erdős and Posa [Canad. J. Math., 17 (1965), pp. 347-352] states that every graph has either k vertex-disjoint cycles or a set of O(k log k) vertices meeting all its cycles. While the standard proof revolves around finding a large "frame" in the graph (a subdivision of a large cubic graph), an alternative way of proving this theorem is to use a ball packing argument of Kuhn and Osthus [Random Structures Algorithms, 22 (2003), pp. 213-225] and Diestel and Rempel [Combinatorica, 25 (2005), pp. 111-116]. In this paper, we argue that the latter approach is particularly well suited for studying edge variants of the Erdős-Pósa theorem. As an illustration, we give a short proof of a theorem of Bruhn, Heinlein, and Joos [Combinatorica, 39 (2019), pp. 1-36] that cycles of length at least ℓ have the so-called edge-Erdős-Pósa property. More precisely, we show that every graph G contains either k edge-disjoint cycles of length at least ℓ or an edge set F of size O(kℓ · log(kℓ)) such that G - F has no cycle of length at least ℓ. For fixed ℓ , this improves on the previously best known bound of O(k2 log k + kℓ). |
