设$A$是一个有理元素的有限矩阵。我们说$A$是{ \it双像划分正则\/}，如果每当正整数的集合${\mathbb N}$是单色的，存在$\vec x$使得$A\vec x$的条目都是相同的颜色（或{ \it单色\/}），并且$\vec x$的条目也是单色的。哪些矩阵是双像分划正则的？更一般地说，我们说一对矩阵$(A,B)$，其中$A$和$B$具有相同的行数，是{ \it双核分区正则\/}，如果每当${\mathbb N}$是单色的，存在矢量$\vec x$和$\vec y$，每个单色，使得$A \vec x + B \vec y = 0$。存在一个正有理$c$，使得矩阵$M=(\begin{array}{ccccc}A&cB\end{array})$是核划分正则的，这是$(A,B)$是二重核划分正则的一个明显的充分条件. (That只要${\mathbb N}$是单色的，就存在单色的$\vec x$，使得$M \vec x=\vec 0$。）本文的目的是证明这个充分条件也是必要的。因此，我们有一个矩阵$A$是双重图像划分正规的当且仅当存在一个正有理$c$，使得矩阵$(\begin{array}{lr}A&cI\end{array})$是核划分正规的，其中$I$是适当大小的单位矩阵。我们还证明了几个矩阵的情况下的扩展。
英文摘要：
Let $A$ be a finite matrix with rational entries. We say that $A$ is {\it doubly image partition regular\/} if whenever the set ${\mathbb N}$ of positive integers is finitely coloured, there exists $\vec x$ such that the entries of $A\vec x$ are all the same colour (or {\it monochromatic\/}) and also, the entries of $\vec x$ are monochromatic. Which matrices are doubly image partition regular?
More generally, we say that a pair of matrices $(A,B)$, where $A$ and $B$ have the same number of rows, is {\it doubly kernel partition regular\/} if whenever ${\mathbb N}$ is finitely coloured, there exist vectors $\vec x$ and $\vec y$, each monochromatic, such that $A \vec x + B \vec y = 0$. There is an obvious sufficient condition for the pair $(A,B)$ to be doubly kernel partition regular, namely that there exists a positive rational $c$ such that the matrix $M=(\begin{array}{ccccc}A&cB\end{array})$ is kernel partition regular. (That is, whenever ${\mathbb N}$ is finitely coloured, there exists monochromatic $\vec x$ such that $M \vec x=\vec 0$.) Our aim in this paper is to show that this sufficient condition is also necessary. As a consequence we have that a matrix $A$ is doubly image partition regular if and only if there is a positive rational $c$ such that the matrix $(\begin{array}{lr}A&cI\end{array})$ is kernel partition regular, where $I$ is the identity matrix of the appropriate size.
We also prove extensions to the case of several matrices.
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A \overset{\rightarrow}{x}$ are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition r…
Image partition regular matrices near zero generalizes many classical results of Ram- sey Theory. There are several characterizations of finite image partition regular matrices near zero. Contrast to the finite cases there are only few classes of matri- ces that are known to be infinite image partition regular near zero. In this present work we have produced several new examples of such classes.
A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $ν_u$ of the Perron eigenvector $\mathbfν$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity …
An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the $5$-regular case and show that among the subgraphs of minimum degree at least $3$, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all $5$-regular Abe…
In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of Glaisher's bijection, the map may be widely useful in other partition combinatorics. A previous conjecture concerning iterations of Glaisher's bijection is given a counterexample.
It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, we propose a method to obtain all the spectrum, and also the local spectra, of a graph $Γ$ from the quotient matrices of some of its regular partitions. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.
We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct' proof of Rado's result.
We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and generating functions: more results for those simultaneously regular and distinct, fewest for those distinct and flat. We offer some conjectures in the area.
Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = Ω(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = Ω(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for b…
For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$, and consider the partition function $M_{G}(λ)=\sum_{k=0}^nm_k(G)λ^k$. In this paper we show that if $G$ is a $d$--regular graph and $0<λ<(4d)^{-2}$, then $$\frac{1}{v(G)}\ln M_G(λ)>\frac{1}{v(K_{d+1})}\ln M_{K_{d+1}}(λ).$$ The same inequality holds true if $d=3$ and $λ<0.3575$. More precise conjectures are also given.
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs
School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
We develop a theory of k-partitions of the set of infinite words recognizable by classes of finite automata. The theory enables to complete proofs of existing results about topological classifications of the (aperiodic) omega-regular k-partitions and provides tools for dealing with other similar questions. In particular, we characterize the structure of Wadge degrees of (aperiodic) omega-regular $k$-partitions, prove the decidability of many related problems, and discuss their complexity.
We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-reg…
