Duality theorems for stars and combs I: Arbitrary stars and combs

Duality theorems for stars and combs II: Dominating stars and dominated combs

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the second paper of the series, we present duality theorems for combinations of stars and combs: dominating stars and dominated combs. As dominating stars exist if and only if dominated combs do, the structures complementary to them coincide. Like for arbitrary stars and combs, our duality theorems for dominated combs (and dominating stars) are phrased in terms of normal trees or tree-decompositions. The complementary structures we provide for …

Duality theorems for stars and combs IV: Undominating stars

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. In the first paper of our series we determined structures that are complementary to arbitrary stars or combs. Stars and combs can be combined, positively as well as negatively. In the second and third paper of our series we provided duality theorems for all but one of the possible combinations. In this fourth and final paper of our series, we complete our solution to the problem of finding complementary structures for stars, combs, and their combination…

非负系数多项式根的Gale对偶界

Gale duality bounds for roots of polynomials with nonnegative coefficients

我们绑定的多项式的根的位置，有非负系数的多项式的次数至多$d$的向量空间的一个固定的，但任意的基础。为此，我们将基多项式解释为实平面上的向量场，并在平面上的每一点上分析盖尔对偶向量配置的组合学。这种方法使我们能够将任意的线性方程和不等式之间的系数在一个统一的方式，以获得更精确的边界上的位置的根。我们应用我们的技术来界定的Ehrhart和色多项式的根的位置。最后，我们给出了一个解释的聚类图中看到的随机多项式的根。

Duality theorems for stars and combs III: Undominated combs

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the third paper of the series, we present duality theorems for a combination of stars and combs: undominated combs. We describe their complementary structures in terms of rayless trees and of tree-decompositions. Applications include a complete characterisation, in terms of normal spanning trees, of the graphs whose rays are dominated but which have no rayless spanning tree. Only two such graphs had so far been constructed, by Seymour and Thoma…

图与拟阵宽度参数的统一对偶定理。II。一般对偶性

Unifying duality theorems for width parameters in graphs and matroids. II. General duality

我们证明了一个一般的对偶定理的纠缠状稠密对象的组合结构，如图和拟阵。本文继续[6]中提出的理论，并假定对该理论熟悉。

Parallel edges in ribbon graphs and interpolating behavior of partial-duality polynomials

Recently, Gross, Mansour and Tucker introduced the partial-twuality polynomials. In this paper, we find that when there are enough parallel edges, any multiple graph is a negative answer to the problem 8.7 in their paper [European J. Combin. 95 (2021), 103329]: Is the restricted-orientable partial-Petrial polynomial of an arbitrary ribbon graph even-interpolating? In addition, we also find a counterexample to the conjecture 8.1 of Gross, Mansour and Tucker: If the partial-dual genus polynomial is neither an odd nor an even polynomial, then it is interpolating.