[论文解读] A new Tutte polynomial for signed graphs

Andrew Goodall, Bart Litjens+2 more|arXiv (Cornell University)|Mar 18, 2019

Advanced Graph Theory Research被引用 1

一句话总结

本文提出了一种签名图的三元 Tutte 多项式，通过引入顶点切换不变性，将经典 Tutte 多项式进行推广，统一了正常着色与无零流的计数。该多项式为同一基集上的拟阵对提供了规范扩展，其独特之处在于能正确捕捉签名图中的流计数，区别于现有双变量色多项式。

ABSTRACT

We introduce the ``trivariate Tutte of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the ``dichromate for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.

研究动机与目标

• 开发一种签名图的不变量，该不变量在顶点切换下保持不变，并推广经典 Tutte 多项式。
• 在单一多项式框架内统一签名图中正常着色与无零流的计数。
• 解决双变量色多项式存在的局限性，即其无法正确捕捉签名图中无零流的数量。
• 将三元 Tutte 多项式扩展至同一基集上的拟阵对，建立规范形式。

提出的方法

• 将签名图的三元 Tutte 多项式定义为边子集上的生成函数，整合平衡性、方向性和切换不变性。
• 使用底层图的环拟阵与签名图的框架拟阵作为同一基集上的拟阵对。
• 通过利用签名图的对称性与拟阵对偶性，建立多项式在顶点切换下的不变性。
• 通过特定参数设置推导多项式的取值，从而获得正常着色数与无零流数。
• 将多项式与 Welsh 和 Kayibi 定义的拟阵对四变量链接多项式相联系，该定义后由 Krajewski、Moffatt 和 Tanasa 进一步形式化。
• 证明与色多项式不同，三元 Tutte 多项式能正确计算签名图中无零张力与无零流的数量。

实验结果

研究问题

• RQ1 如何构建一种签名图的 Tutte 型多项式，使其在顶点切换下保持不变，同时捕捉着色与流？
• RQ2 为何双变量色多项式无法枚举签名图中的无零流，如何修正这一问题？
• RQ3 Tutte 多项式对同一基集上拟阵对的规范扩展是什么，其与签名图有何关联？
• RQ4 签名图的三元 Tutte 多项式与 Welsh 和 Kayibi 定义的拟阵对链接多项式之间有何关系？
• RQ5 三元 Tutte 多项式能否用于计算签名图中无零张力的数量，若可以，应在哪些参数取值下进行？

主要发现

• 签名图的三元 Tutte 多项式在顶点切换下保持不变，并推广了经典 Tutte 多项式。
• 签名图的正常着色数可通过三元 Tutte 多项式在特定参数值下的特化获得。
• 无零流的数量通过多项式在两个不同点的取值被捕捉，证明了其在流计数中的作用。
• 三元 Tutte 多项式与双变量色多项式有本质区别，后者无法正确给出签名图中无零流的计数。
• 该多项式可规范地扩展为同一基集上拟阵对的不变量，与 Krajewski、Moffatt 和 Tanasa 的四变量链接多项式一致。
• 通过特定取值，三元 Tutte 多项式能正确计算签名图中无零张力的数量，解决了先前方法的关键局限。