关于惠特尼断回路定理的一个注记

Bijective proofs of proper coloring theorems

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynom…

Recursion relations for chromatic coefficients for graphs and hypergraphs

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.

An inequality on broken chessboards

For any partition of a positive integer we consider the chess (or draughts) colouring of its associated Ferrers graph. Let b denote the total number of black unit squares, and w the number of white squares. In this note we characterize all pairs (b,w) which arise in this way. This simple combinatorical result was discovered by characterizing Hilbert series of certain right modules over cubic three-dimensional Artin-Schelter algebras. However in this note we present a purely combinatorical proof. The result is (at least partially) known in literature, however we found it interesting to present …

半拟阵的特征多项式及其与拟阵、超平面构形和图着色的联系

Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings

该研究探究半拟阵特征多项式的性质，推广惠特尼断链定理与罗塔-赫伦-韦尔奇猜想，给出相关卷积恒等式，并引入指派拟阵建立半拟阵、超平面构形与图着色的联系。

Dominic Welsh：他的工作与影响

Dominic Welsh: his work and influence

本文回顾了Dominic Welsh在离散数学领域的广泛贡献，重点强调其在拟阵理论和Tutte-Whitney多项式方面的工作及深远影响。

Flow polynomials of a signed graph

In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown that there exists a polynomial $F_d(G,x)$ such that the number of the nowhere-zero $Γ$-flows in $G$ equals $F_d(G,x)$ evaluated at $k$ for every Abelian group $Γ$ of order $k$ with $ε(Γ)=d$, where $ε(Γ)$ is the largest integer $d$ for which $Γ$ has a subgroup isomorphic to $\mathbb{Z}^d_2$. We focus on the combinatorial structure of $Γ$-flows in a signed gra…