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تسجيل مستخدم جديدHalf-integral ErdH{o}s-Posa property of directed odd cycles
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We prove that there exists a function $f:mathbb{N}rightarrow mathbb{R}$ such that every digraph $G$ contains either $k$ directed odd cycles where every vertex of $G$ is contained in at most two of them, or a vertex set $X$ of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral ErdH{o}s-Posa property of undirected odd cycles, proved by Reed [Mangoes and blueberries. Combinatorica 1999], to digraphs.
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A chordless cycle, or equivalently a hole, in a graph $G$ is an induced subgraph of $G$ which is a cycle of length at least $4$. We prove that the ErdH{o}s-Posa property holds for chordless cycles, which resolves the major open question concerning th
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Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the ErdH{o}s-Posa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the ErdH{o}s-Posa property, an
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We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such
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A Group Labeled Graph is a pair $(G,Lambda)$ where $G$ is an oriented graph and $Lambda$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$
, the group element obtained by `adding the labels on forward arcs and `subtracting the labels on reverse arcs is not the identity element of the group. Non-null cycles in group labeled graphs generalize several well-known graph structures, including odd cycles. In this paper, we prove that non-null cycles on Group Labeled Graphs have the half-integral Erdos-Posa property. That is, there is a function $f:{mathbb N}to {mathbb N}$ such that for any $kin {mathbb N}$, any group labeled graph $(G,Lambda)$ has a set of $k$ non-null cycles such that each vertex of $G$ appears in at most two of these cycles or there is a set of at most $f(k)$ vertices that intersects every non-null cycle. Since it is known that non-null cycles do not have the integeral Erdos-Posa property in general, a half-integral Erdos-Posa result is the best one could hope for.
ErdH{o}s and P{o}sa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed prov
ed an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.
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