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Erdös--Pósa from Ball Packing
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-07-27 , DOI: 10.1137/19m1309225 Wouter Cames van Batenburg , Gwenaël Joret , Arthur Ulmer
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-07-27 , DOI: 10.1137/19m1309225 Wouter Cames van Batenburg , Gwenaël Joret , Arthur Ulmer
SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1609-1619, January 2020.
A classic theorem of Erdös and Pósa [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 Kühn 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 Erdös--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 $\ell$ have the so-called edge-Erdös--Pósa property. More precisely, we show that every graph $G$ contains either $k$ edge-disjoint cycles of length at least $\ell$ or an edge set $F$ of size $O(k\ell \cdot \log (k\ell))$ such that $G-F$ has no cycle of length at least $\ell$. For fixed $\ell$, this improves on the previously best known bound of $O(k^2 \log k +k\ell)$.
中文翻译:
球包装的Erdös-Pósa
SIAM离散数学杂志,第34卷,第3期,第1609-1619页,2020年1月。
Erdös和Pósa[Canad。J. Math。,17(1965),pp。347--352]指出,每个图都有$ k $个顶点不相交的周期或一组满足所有周期的$ O(k \ log k)$个顶点。虽然标准证明是围绕在图中找到一个大“框架”(一个大三次方图的一个细分)而进行的,但证明该定理的另一种方法是使用Kühn和Osthus的球堆积参数[Random Structures Algorithms,22( (2003年),第213--225页和Diestel和Rempel [Combinatorica,25(2005),第111--116页]。在本文中,我们认为后一种方法特别适合研究Erdös-Pósa定理的边变体。作为说明,我们给出了Bruhn,Heinlein和Joos定理的简短证明[Combinatorica,39(2019),pp。[1--36]长度至少为$ \ ell $的周期具有所谓的edge-Erdös-Pósa属性。更确切地说,我们显示每个图$ G $包含长度至少为$ \ ell $的$ k $边不相交周期或大小为O的边集$ F $(k \ ell \ cdot \ log(k \ ell))$,这样$ GF $就没有长度循环,至少$ \ ell $。对于固定的$ \ ell $,这会改善先前最广为人知的$ O(k ^ 2 \ log k + k \ ell)$的边界。
更新日期:2020-07-27
A classic theorem of Erdös and Pósa [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 Kühn 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 Erdös--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 $\ell$ have the so-called edge-Erdös--Pósa property. More precisely, we show that every graph $G$ contains either $k$ edge-disjoint cycles of length at least $\ell$ or an edge set $F$ of size $O(k\ell \cdot \log (k\ell))$ such that $G-F$ has no cycle of length at least $\ell$. For fixed $\ell$, this improves on the previously best known bound of $O(k^2 \log k +k\ell)$.
中文翻译:
球包装的Erdös-Pósa
SIAM离散数学杂志,第34卷,第3期,第1609-1619页,2020年1月。
Erdös和Pósa[Canad。J. Math。,17(1965),pp。347--352]指出,每个图都有$ k $个顶点不相交的周期或一组满足所有周期的$ O(k \ log k)$个顶点。虽然标准证明是围绕在图中找到一个大“框架”(一个大三次方图的一个细分)而进行的,但证明该定理的另一种方法是使用Kühn和Osthus的球堆积参数[Random Structures Algorithms,22( (2003年),第213--225页和Diestel和Rempel [Combinatorica,25(2005),第111--116页]。在本文中,我们认为后一种方法特别适合研究Erdös-Pósa定理的边变体。作为说明,我们给出了Bruhn,Heinlein和Joos定理的简短证明[Combinatorica,39(2019),pp。[1--36]长度至少为$ \ ell $的周期具有所谓的edge-Erdös-Pósa属性。更确切地说,我们显示每个图$ G $包含长度至少为$ \ ell $的$ k $边不相交周期或大小为O的边集$ F $(k \ ell \ cdot \ log(k \ ell))$,这样$ GF $就没有长度循环,至少$ \ ell $。对于固定的$ \ ell $,这会改善先前最广为人知的$ O(k ^ 2 \ log k + k \ ell)$的边界。