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引用次数: 0
摘要
Kochol [6]利用兼容集的概念给出了矩阵的 Tutte 多项式的新展开式,并提出了这一展开式与内部-外部活动式之间的关系。在这里,我们给出了答案,它是 Las Vergnas 的矩阵视角三变量 Tutte 多项式的扩展公式的广义版本的特例[10]。与这项工作平行,Kochol 在 [5] 和 [7] 中独立证明了对 matroid 透视图的相同广义化和与活动的双射,但使用的方法不同。Kochol 利用收缩-删除关系递归证明了这两个结果,而我们则更直接地证明了双射,并利用双射从 Las Vergnas 的活动展开推导出了兼容集展开公式。
On the Compatible Sets Expansion of the Tutte Polynomial
Kochol [6] gave a new expansion formula for the Tutte polynomial of a matroid using the notion of compatible sets, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas’s trivariate Tutte polynomials of matroid perspectives [10]. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol in [5] and [7] in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas’s activities expansion.
期刊介绍:
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches