Jan Hladký, Daniel Král', Uwe Schauz
We give a proof of Brooks' theorem and its list coloring extension using the algebraic method of Alon and Tarsi; this also shows that the Brooks' theorem remains valid in a more general game coloring setting.
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We present a short and self-contained proof of the choosability version of Brooks' theorem.
We give a new proof of Brooks' theorem that immediately implies a strengthening of Brooks' theorem, known as Catlin's theorem.
We give a simple short proof of Brooks' theorem using only induction and greedy coloring, while avoiding issues of graph connectivity. The argument generalizes easily to some extensions of Brooks' theorem, including its variants for list coloring, signed graphs coloring and correspondence coloring.
Lovász gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.
Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.
Four proofs of the directed Brooks' Theorem
本文针对有向 Brooks 定理给出四个新证明，并补充了一个 NP 完全性结果。
We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon--Tarsi orientations, since analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin--Kostochka Conjecture and Reed's Conjecture.
On Brooks' Theorem
本研究针对Brooks定理，通过修改已有证明、结合两种已有证明的方式提出两种更易在计算机科学课程中教学的证明方法。
Dirac's theorem on chordal graphs implies Brooks' theorem
本文利用狄拉克关于弦图的著名定理，为布鲁克斯定理的列表着色版本提供了一个新的证明。
Greedy Coloring and Simple Exponential Strengthenings of Brooks-Type Theorems
本文对布鲁克斯定理给出新证明，证明连通非完全且最大度Δ≥3的图存在指数多个正常Δ着色，还拓展技术到多种着色场景，给出新归约与度可选性定理的新证明。
Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let $D\geq 3$ and $G$ be a graph without cliques on $D+1$ vertices and the degree of any vertex in this graph is not greater than $D$. Then for every integer $c\geq 2$ there is a proper $(c,p)$-nondegenerate vertex $D$-coloring of $G$, where $p=(c^3+8c^2+19c+6)(c+1).$ During t…
We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $μ$-measurable $d$-coloring with respect to any Borel probability measure $μ$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of …
Dirac introduced the notion of a k-critical graph, a graph that is not (k-1)-colorable but whose every proper subgraph is (k-1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a subgraph isomorphic to K_{k+1} (or an odd cycle for k=2). Equivalently, for all k>=4, the only k-critical graph of maximum degree k-1 is K_k. A natural generalization of Brook's theorem is to consider the Ore-degree of a graph, which is the maximum of d(u)+d(v) over all edges uv. Kierstead and Kostochka proved that for all k>=6 the only k-critical graph with Or…
In 1998 the second author proved that there is an $ε>0$ such that every graph satisfies $χ\leq \lceil (1-ε)(Δ+1)+εω\rceil$. The first author recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Hall's Theorem, Brooks' Theorem, the Lovász Local Lemma, and Talagrand's Inequality.
Short Proofs in Algebraic and Enumerative Combinatorics
本文利用ChatGPT 5.4 Pro自主生成简短证明，解决了模格上的梯队运动算子、停车函数统计以及plactic幺半群中心化子等领域的若干开放问题。
An algebraic-combinatorial proof of a Bezout-type inequality for mixed volumes of three-dimensional zonoids
本文用新的代数组合方法重新证明了三维环带体混合体积的Bezout型不等式，并揭示了该不等式与实代数和拟阵论之间的深层联系。
In this paper we give a proof that the largest set of perfect matchings, in which any two contain a common edge, is the set of all perfect matchings that contain a fixed edge. This is a version of the famous Erdős-Ko-Rado theorem for perfect matchings. The proof given in this paper is algebraic, we first determine the least eigenvalue of the perfect matching derangement graph and use properties of the perfect matching polytope. We also prove that the perfect matching derangement graph is not a Cayley graph.
An Algebraic Proof of the Polynomial Version of van der Waerden's Theorem
针对范德瓦尔登定理的多项式版本，本文引入符号多项式，给出了一种仅基于无限离散半群Stone-Čech紧化的替代代数证明，为该定理提供了新的证明路径。
I. P. Goulden, S. Litsyn, and V. Shevelev [On a sequence arising in algebraic geometry, J. Integer Sequences 8 (2005), 05.4.7] conjectured that certain Laurent polynomials associated with the solution of a functional equation have only odd negative powers. We prove their conjecture and generalize it.
A linear algebraic proof of the Laplacian spread conjecture
研究拉普拉斯谱展猜想，即\(\alpha(G)+\alpha(\overline{G}) \geq 1\)，核心方法是线性代数，主要贡献是给出更简洁的新证明。
Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov
本文针对图论领域，证明了Kolokolnikov于2015年提出的关于特定阶数和边数的图的最大代数连通性的猜想。
We give an algebraic proof of the determinant formulas for factorial Grothendieck polynomials obtained by Hudson--Ikeda--Matsumura--Naruse and by Hudson--Matsumura.
Towards inductive proofs in algebraic combinatorics
本文引入了一类新的传递置换群，给出了其商保持在该族中的充分条件，并利用归纳法确定了阶为$p^n$的循环有向图的自同构群。
We consider planar lattice walks that start from (0,0), remain inthe first quadrant i, j >= 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: -- they are counted by nice numbers (Kreweras 1965), -- the generating function of these numbers is algebraic (Gessel 1986), -- the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function (Flatto and Hahn 1984). These results are not well understood, and have been established via complica…
We prove bounds on the number of incidences between a set of algebraic curves in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz.
We give simpler algebraic proofs of uniqueness for several Erdős-Ko-Rado results, i.e., that the canonically intersecting families are the only largest intersecting families. Using these techniques, we characterize the largest partially 2-intersecting families of perfect hypermatchings, resolving a recent conjecture of Meagher, Shirazi, and Stevens.
Formalising Fisher's Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics
本文提出了在Isabelle/HOL中用于关联结构证明的形式化线性代数技术，首次完成了Fisher不等式的形式化，并展示了该技术可适应组合学的未来形式化工作。
For a graph $G$, let $λ_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $λ_2(G) + λ_2(\overline{G}) \geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this conjecture in the general case. Also, we will show that $\max\{λ_2(G), λ_2(\overline{G})\} \geq 1 - O(n^{-\frac 13})$, where $n$ is the number of vertices of $G$.
Combinatorial and algebraic proofs of Keller's $A_2\square A_2$ $q$-dilogarithm identity
该研究针对与平方积箭图$A_2\square A_2$相关的凯勒$q$-双对数恒等式，通过生成函数、带色分拆双射、四色弧代数三种方法，以迭代Durfee分解为核心完成了组合与代数证明，补充了该恒等式的新证明途径。
In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in $\mathbb{R}^n$. If $n$ is odd, then there are two linked $(n+1)/2$-simplices with the vertices at these points. If $n$ is even, then one can choose two disjoint $(n+2)/2$-tuples such that the interiors $(n/2)$-simplices with the vertices at these $(n+2)/2$-tuples intersect each other.} This theorem is interesting even in case of small dimensions.
Machine Checked Proofs and Programs in Algebraic Combinatorics
本文介绍一个基于 Coq/Rocq 与 Mathematical Components 的代数组合学形式化库，核心贡献是机器检验的 Littlewood-Richardson 法则证明及相关认证算法实现。
An algebraic proof of the dichotomy for graph orientation problems with forbidden tournaments
本研究利用光滑逼近理论，给出了带禁止竞赛图的无F定向问题复杂性二分性的更短新证明，还将结果推广至带额外局部约束的更广图定向问题类，且其难易边界可由可判定代数条件描述。
The classic Cayley identity states that \det(\partial) (\det X)^s = s(s+1)...(s+n-1) (\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \partial=(\partial/\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-par…
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