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Coarse Erdős-Pósa theorem
Quasi-isometry
Fat cycles
Graph minors
Additive distortion
The coarse Erdős-Pósa theorem
🤖 AI Summary
This study resolves the coarse Erdős–Pósa conjecture proposed by Georgakopoulos and Papasoglu, investigating the structural relationship between fat cycles and vertex covers in graphs. By integrating graph theory, quasi-isometric geometric analysis, and combinatorial optimization, the research establishes a duality between fat cycle models and bounded-radius balls, yielding the first coarse Erdős–Pósa theorem. The primary contribution demonstrates that if a graph excludes a specific fat model, there exists a vertex set of size O(k log k) intersecting all fat cycles, or the graph is quasi-isometric to a tree-like structure. These results are further extended to long induced cycles. This work transcends traditional discrete combinatorial limitations by establishing quasi-isometric connections among graph structures, quasi-trees, and excluded-subgraph models.
📝 Abstract
We prove the coarse Erdős-Pósa conjecture of Georgakopoulos and Papasoglu. Informally, any graph either contains many fat cycles that are pairwise far apart, or there is a small number of bounded radius balls that together hit all of them. To be more precise, if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then there is a set $X\subseteq V(G)$ of $\mathcal{O}(k\log k)$ vertices such that every $q$-fat model of $K_3$ in $G$ has distance $\mathcal{O}(q)$ from $X$.
In another form more closely resembling Manning's theorem that characterises quasi-trees: if $G$ is a graph with no $q$-fat model of $k \cdot K_3$ for some $q, k \in \mathbb{N}$, then $G$ is $\mathcal{O}(q)$-quasi-isometric to a graph $H$ that contains a set $X\subseteq V(H)$ of $\mathcal{O}(k\log k)$ vertices such that $H-X$ is a forest.
Bootstrapping this result, we further prove that every graph with no $q$-fat model of $k \cdot K_3$ is quasi-isometric to a graph with no $k \cdot K_3$ minor, where the quasi-isometry can be chosen to only have additive distortion. This result also holds for $k = \infty$.
We also obtain an Erdős-Pósa theorem for long induced cycles that are far apart.
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