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تسجيل مستخدم جديدThe edge-ErdH{o}s-Posa property
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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, and indeed even fails when $H$ is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.
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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
e ErdH{o}s-Posa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either $k+1$ vertex-disjoint chordless cycles, or $c_1k^2 log k+c_2$ vertices hitting every chordless cycle for some constants $c_1$ and $c_2$. It immediately implies an approximation algorithm of factor $mathcal{O}(sf{opt}log {sf opt})$ for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least $ell$ for any fixed $ellge 5$ do not have the ErdH{o}s-Posa property.
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 $f(k)$ hitt
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Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t geq 3$ there is a constant $c=c(t)$ such that for every integer $kgeq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor
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Let $mathbf{k} := (k_1,dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ cont
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