[论文解读] Companion to \An update on the Hirsch conjecture"

Edward D. KimFrancisco Santos|arXiv (Cornell University)|Jan 1, 2010

Advanced Combinatorial Mathematics参考文献 57被引用 4

一句话总结

本篇配套论文为《Hirsch猜想更新》主论文中省略的选定结果提供了完整的证明和额外评述。聚焦于多面体组合学中的Hirsch猜想，本文通过涉及单纯多面体和脊路的对偶方法，确立了三维中n个面的d维多面体的直径的精确界限H(n,3) = ⌊2n/3⌋ − 1。主要贡献是通过三维多面体中的分层顶点构型，构造性地证明了该紧致界限。

ABSTRACT

AbstractThis is an appendix to our paper \An update of the Hirsch Conjec-ture, containing proofs of some of the results and comments that wereomitted in it. 1 Introduction This is an appendix to our paper \An update of the Hirsch Conjecture [39],containing proofs of some of the results and comments that were omitted in it.The numbering of sections and results is the same in both papers, although notall appear in this companion. The same occurs with the bibliography, whichwe repeat here completely although not all of the papers are referenced. Thenumbering of gures, however, is correlative. Figures 1 to 6 are in [39] andFigures 7 to 16 are here. 2 Bounds and algorithms 2.1 Small dimension or few facets *Theorem 2.1 (Klee [40]). H(n;3) = b 2n3 c 1.Proof. To prove the lower bound, we work in the dual setting where our polytopeP simplicial and we want to move from one facet to another along the ridgesof P. Figure 7 shows the graph of a simplicial 3-polytope with nine vertices inwhich ve steps are needed to go from the interior triangle to the most externalone (the outer face in the picture, which represents a facet in the polytope).The reader can easily generalize the gure to any number of vertices divisibleby three, adding layers of three vertices that increase the diameter by two.For a number of vertices equal to one or two modulo three, simply add one ortwo vertices in the interior of the central triangle, subdividing it into three or

研究动机与目标

• 为在主论文中省略的Hirsch猜想相关结果提供完整证明。
• 确立n个面的三维多面体的直径界限H(n,3) = ⌊2n/3⌋ − 1的精确值。
• 提供几何构造方法，并通过单纯多面体和脊路路径提供对偶解释。
• 澄清并扩展关于三维多面体中顶点层与面连通性的组合论证。

提出的方法

• 采用对偶设定，其中多面体为单纯形，通过沿面之间的脊路分析直径。
• 通过每层包含三个顶点的分层集合构造三维多面体，使直径每层增加2。
• 对于n ≡ 1或2 (mod 3) 的情况，通过在中心三角形内部添加一个或两个顶点以实现适当细分。
• 应用Klee的结果（定理2.1），通过组合与几何推理推导出精确界限H(n,3) = ⌊2n/3⌋ − 1。
• 使用图示（图7–16）说明分层构型及对偶图中的脊路路径。

实验结果

研究问题

• RQ1 n个面的三维多面体的直径精确值是多少，如何实现紧致界定？
• RQ2 如何通过单纯多面体的对偶方法，在三维这一特殊情形下证明Hirsch猜想？
• RQ3 何种几何构造可实现每层三个顶点系统性地使直径增加2？
• RQ4 在非可被3整除的情形（n ≡ 1或2 mod 3）下，向中心三角形内部添加顶点如何影响直径？

主要发现

• n个面的三维多面体的直径精确为H(n,3) = ⌊2n/3⌋ − 1。
• 通过每组三个顶点的分层构造，三维多面体的直径每层增加2。
• 当n ≡ 1或2 (mod 3) 时，通过在中心三角形内部添加一个或两个顶点，可实现适当细分并维持直径界限。
• 采用单纯多面体与脊路的对偶方法，为该界限提供了几何上直观的证明。