彩色子式的Erdős-Pósa性质
The Erdős-Pósa Property for Colorful Minors
AI总结：
该研究确定了具有彩色子式关系的Erdős-Pósa性质的彩色图，通过结构形式、障碍集、网格形式三种等价形式完成刻画，拓展了经典的Erdős-Pósa性质理论。
AI中文摘要：
彩色图关系通过在收缩时合并颜色集并允许移除颜色来增强子式关系，它推广了有根子式并对具有多个可能重叠的标注顶点集的图上问题进行建模。根据Robertson和Seymour的经典定理，一个图具有子式的Erdős-Pósa性质当且仅当它是平面图。在这项工作中，我们针对彩色子式关系确定了哪些彩色图具有Erdős-Pósa性质。我们的刻画采用三种等价形式：第一种是结构形式，具有该性质的彩色图是那些可以将所有带色顶点绘制在一个面上，且其颜色以精确意义沿该面排布而不交错的图；第二种由障碍集给出，它们是那些排除了显式无限族$\boldsymbol{\textit{O}}$中每个成员的图，对于每个有限颜色集$I$，其中仅$\boldsymbol{O}(|I|^{4})$个成员的颜色是$I$的子集；第三种是网格形式，它们恰好是特定分离网格族的并集的彩色子式，这些分离网格是驱动经典证明的网格的彩色类似物。
英文摘要：
A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\mathcal{O},$ of which only $\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.
