[论文解读] Topological lower bounds for the chromatic number: Ah ierarchy

Martin Ziegler|arXiv (Cornell University)|Jan 1, 2003

Topological and Geometric Data Analysis参考文献 32被引用 80

一句话总结

本文建立了一个图的色数的拓扑下界层次结构，表明洛瓦兹原始的邻域复形下界是最强的。文章引入了一种简化且函子式的方盒复形构造，该构造与洛瓦兹的复形等价，统一并强化了通过Borsuk–Ulam型定理估计色数的现有拓扑方法。

ABSTRACT

This paper is a study of topological lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovasz in 1978, in his famous proof of the Kneser conjecture via Algebraic Topology. This conjecture stated that the Kneser graph KGm,n ,t he graph with allk-element subsets of {1,2,...,n} as vertices and all pairs of disjoint sets as edges, has chromatic number n! 2k+2. Several other proofs have since been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Kr´iz, Greene, and others), all of them based on some version of the Borsuk-Ulam theorem, but otherwise quite di!erent. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe that every finite graph may be represented as a generalized Kneser graph, to which the above bounds apply.) We show that these bounds are almost linearly ordered by strength, the strongest one being essentially Lovasz' original bound in terms of a neighborhood complex. We also present and compare various definitions of a box complex of a graph (developing ideas of Alon, Frankl, and Lovasz and of Kr´iz). A suitable box complex is equivalent to Lovasz' complex, but the construction is simpler and functorial, mapping graphs with homomorphisms to Z2-spaces with Z2-maps.

研究动机与目标

• 系统比较并排名有限图色数的拓扑下界。
• 澄清并统一拓扑图论中使用的方盒复形的定义。
• 证明所有有限图均可表示为广义Kneser图，从而可应用拓扑下界。
• 表明洛瓦兹原始的邻域复形下界在该类下界层次结构中最强。
• 开发一种更简单、函子式的方盒复形构造，能将图同态映射为Z2-空间之间的Z2-映射。

提出的方法

• 构建一种新的方盒复形，其与洛瓦兹的邻域复形等价，但定义更简单且更具函子性。
• 以Borsuk–Ulam定理作为基础工具，推导色数的拓扑下界。
• 定义广义Kneser图，以证明每个有限图均可作为此类图出现，从而将拓扑下界推广至所有图。
• 比较Alon、Frankl、Lovász和Kríz所提出的方盒复形的不同定义，识别出等价与更强的表述。
• 通过分析Z2作用下其拓扑不变量的强度，建立下界层次结构。
• 证明邻域复形下界在该层次结构中主导所有其他已知的拓扑下界。

实验结果

研究问题

• RQ1 不同色数的拓扑下界在强度上如何比较？
• RQ2 能否定义一种统一的、函子式的方盒复形构造，以保持图同态的性质？
• RQ3 洛瓦兹的邻域复形下界是否是所有已知拓扑下界中最强大的？
• RQ4 是否每个有限图都可嵌入为广义Kneser图，以应用拓扑下界？
• RQ5 文献中各种方盒复形定义之间的关系是什么？

主要发现

• 洛瓦兹原始的邻域复形下界是色数拓扑下界层次结构中最强的。
• 一种新的、更简单的方盒复形构造与洛瓦兹的邻域复形等价，且具有函子性，能将图同态映射为Z2-空间之间的Z2-映射。
• 所有有限图均可表示为广义Kneser图，从而可普遍应用拓扑下界。
• 拓扑下界层次结构几乎是线性有序的，仅存在少数强度上的例外。
• 方盒复形构造简化并统一了先前的方法，特别是Alon、Frankl、Lovász和Kríz的研究。
• 新方盒复形与洛瓦兹复形的等价性证实了色数估计拓扑方法的稳健性。