发表机构
University of Manchester（曼彻斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明对任意k≥3，任意有限丛可由k-坏奇圈的极小终端迹实现，保持打包-覆盖数，导致有限拥塞和分数Erdős-Pósa性质失效，与k=2情形形成对比。
AI 中文摘要
对于每个固定的$k\ge 3$，每个有限非空丛$\mathcal{H}$都可以精确实现为$k$-坏奇圈（即满足$\chi(G[V(C)])\ge k+1$的奇圈$C$）的包含极小终端迹族。宿主图$G$可选择为$2$-连通的，且$T=V(\mathcal{H})$独立，并满足$\chi(G)=k+1$，$\omega(G)=k$，以及$|E(G)|\le k|V(G)|$。给定任意整数$B\ge 1$，一个实现同时保持横贯数、分数打包数以及所有$1\le c\le B$的整数$c$-打包数。因此，奇圈跨上的色丰富性具有任意有限集合系统的完整打包-覆盖复杂度，即使在色数比团数大一的稀疏图中也是如此。作为推论，对于每个$b,N\ge 1$和每个$\varepsilon>0$，存在这样的图，使得$\tau_k(G)=N$，对所有$1\le c\le b$有$\nu_k^{1/c}(G)=c$，且$\nu_k^*(G)<1+\varepsilon$。因此，对于$k\ge 3$，每个有限拥塞和分数Erdős--Pósa性质均失效，并具有渐近最优的分数障碍。这与$k=2$形成鲜明对比，其中相关圈是普通奇圈：整数性质失效，而Reed定理给出至少为2的每个拥塞水平以及分数性质。我们还通过单个固定见证图刻画了最短的$k$-坏奇圈。
英文摘要
For every fixed $k\ge 3$, every finite nonempty clutter $\mathcal{H}$ can be realized exactly as the family of inclusion-minimal terminal traces of the $k$-bad odd cycles, namely the odd cycles $C$ satisfying $χ(G[V(C)])\ge k+1$. The host graph $G$ may be chosen $2$-connected, with $T=V(\mathcal{H})$ independent, and with $χ(G)=k+1$, $ω(G)=k$, and $|E(G)|\le k|V(G)|$. Given any integer $B\ge 1$, one realization simultaneously preserves the transversal number, the fractional packing number, and every integer $c$-packing number for $1\le c\le B$. Thus chromatic richness on odd-cycle spans has the full packing-covering complexity of arbitrary finite set systems, even in sparse graphs whose chromatic number exceeds their clique number by one. As a consequence, for every $b,N\ge 1$ and every $\varepsilon>0$, there is such a graph with $τ_k(G)=N$, $ν_k^{1/c}(G)=c$ for every $1\le c\le b$, and $ν_k^*(G)<1+\varepsilon$. Hence every finite-congestion and fractional Erdős--Pósa property fails for $k\ge 3$, with an asymptotically optimal fractional obstruction. This contrasts sharply with $k=2$, where the relevant cycles are the ordinary odd cycles: the integral property fails, while Reed's theorem yields every congestion level at least two and the fractional property. We also characterize the shortest $k$-bad odd cycles by a single fixed witness graph.
