Donald E. Knuth
A set partition technique that is useful for identifying wires in cables can be recast in the language of 0--1 matrices, thereby resolving an open problem stated by R.~L. Graham in Volume 1 of this journal. The proof involves a construction of 0--1 matrices having row and column sums without gaps.
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Erdős & Graham ask whether the equation $x^2+y^2=z^2$ is partition regular, i.e. whether it has a finite Rado number. This note provides a lower bound and also states results in the affirmative for two similar quadratic equations.
Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that $G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$, $|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of the graph $H$. Little is known about this question apart from a few simple observations. We show that the number of trees on $n$ vertices which can be distinguished by their associated integer sequences is $e^{Ω((\log n)^{3/2})}$. The proof strategy involves constructing a large collection of caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott problem.
For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answ…
A counterexample to the Chung-Graham-Spiro gap-set conjecture
针对Chung-Graham-Spiro间隙集猜想，研究人员通过证明9属于上整数的4步间隙集但不属于下整数的4步间隙集，得出该猜想在ℓ=4时不成立的结论。
We apply the method of characteristics for the solution of pde's to two combinatorial problems. The first is finding an explicit form for a distribution that arises in bio-informatics. The second is a question raised by Graham, Knuth and Patashnik abiout a sequence of generalized binomial coefficients. We find an exact formula, which factors in an interesting way, in the case where one of the six parameters of the problem vanishes. We also show that the associated polynomial sequence has real zeros only, provided that one parameter vanishes, and the other five are nonnegative.
Structural rigidity in the Erdős-Graham two-set permutation problem
本文针对Erdős-Graham两集合置换问题，通过研究二元候选集S_A的结构限制，证明其不存在可允许置换，表明典型二元划分无法解决该问题，该问题仍待解决。
Exact Biclique Partition number of Split Graphs
研究分割图的双 clique 分解数，证明其等于补图最大 clique 数减一，扩展了格雷厄姆-波拉克定理的应用范围。
In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are monochromatic. They give an explicit upper bound: F(F(F(F(F(F(F(12))))))), where F(m) = 2^^(m)^^3, an extremely fast-growing function. By reducing the problem to a variant of the Hales-Jewett problem, we find an upper bound which is between F(4) and F(5).
New bounds on the Graham-Pollak theorem for hypergraphs
本文改进了超图格雷厄姆-波拉克定理的界限，将c_r的最小值从113降低到85，并提高了f_r(n)的界限。
On a Conjecture about Ron Graham's Sequence
本文研究了Ron Graham序列的性质，提出了伪多项式时间计算其值的算法，并证明了关于该函数上界的一个22年历史的猜想。
On the maximal anti-Ramsey problem of Burr, Erdős, Graham, and Sós for $P_4$
研究Burr、Erdős、Graham和Sós关于P₄的最大反拉姆齐问题，Li等人对0<ε<1/2给出否定答案，本文证明在ε≥1/2时存在二次下界χS(n,⌊n²⌋ - ⌊n²⁻ε⌋,P₄)>c n² 。
On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences
本文证明了存在常数c>0，使得对任意正整数n，可选取1到n间的递增序列，其连续子段和的不同值至少为cn^2个，回答了Erdős和Graham的问题，并给出上界及解决若干开放问题。
Proving it is impossible; on Erdős problem $\# 278$
该研究针对埃尔德什问题#278，给出自然结构化族的精确表达式并关联平衡划分实例，证明二进制编码的重复模数列表下判断最小密度是否为零是NP困难的。
We consider the problem of computing the second elementary symmetric polynomial S^2_n(X) using depth-three arithmetic circuits of the form "sum of products of linear forms". We consider this problem over several fields and determine EXACTLY the number of multiplication gates required. The lower bounds are proved for inhomogeneous circuits where the linear forms are allowed to have constants; the upper bounds are proved in the homogeneous model. For reals and rationals, the number of multiplication gates required is exactly n-1; in most other cases, it is \ceil{n/2}. This problem is related to …
Two problems of Burr, Erd\H os, Graham, and Sós on maximal anti-Ramsey functions for $P_4$
本文回答了Burr等人提出的关于$P_4$的最大反拉姆齐函数的两个问题：第一个问题肯定，第二个问题否定，证明分别使用了局部密度不等式和Alon等人的构造。
Cellular subalgebras of the partition algebra
本文研究分区代数等图代数的细胞结构，利用半图分解证明相关图代数均为细胞代数并参数化其细胞模，同时提出从一般细胞代数与 rook Brauer 代数子代数构造新细胞代数的 cellular wreath product 方法。
A counterexample to the conjecture on Biclique Partition number of Split Graphs and related problems
本文通过构造Split图反例，推翻了biclique分解数与补图最大团数关系的猜想，并探讨了Split图biclique分解的结构性质及二元秩问题。
We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration $T$ which is a vertex set of a non-equilateral triangle there is a copy of $T$ contained in the interior of one of the two partition classes. Furthermore, we give the characterization of these "polygonal" p…
We consider the chessboard pebbling problem analyzed by Chung, Graham, Morrison and Odlyzko [3]. We study the number of reachable configurations $G(k)$ and a related double sequence $G(k,m)$. Exact expressions for these are derived, and we then consider various asymptotic limits.
A note on universal graphs for spanning trees
本文针对含所有n顶点树为子图的n顶点通用图的最小边数问题，改进了Chung和Graham给出的1/2 n log n边数下界，将其提升至n log n。
On Graham's rearrangement conjecture over $\mathbb{F}_2^n$
本文研究群元素子集的有效排序问题，针对G=F_2^n情形给出本质完全解答：证明存在绝对常数C，使所有大小至少为C的子集均可有效排序；证明结合Freiman–Ruzsa定理与吸收方法，并借助Cayley图的拟随机分解结构。
A resolution of Erdős Problem #190 via Erdős-Lovász, BCT, and Baker-Harman-Pintz
本文利用Erdős-Lovász、BCT和Baker-Harman-Pintz定理，证明了H(k)^{1/k}/k随着k增长而趋于无穷，解决了Erdős-Graham问题的正向方向。
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.
The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (pa…
On a specific problem of partition of graph
本文研究图顶点的2-划分问题，证明当图可划分为若干“小”图时，任意割的跨边数$e(S,S^{c})$均存在良好下界。
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially ma…
A Finite-Bound Partition Equinumerosity Result Generalizing a Solution of a Problem Posed by Andrews and Deutsch
本研究对Andrews、Deutsch2016年提出、后经Smoot和Yang2018年推广的分拆等计数结论做有限界拓展，并给出对应双射证明。
Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we g…
For $n$ and $k$ integers we introduce the notion of some partition of $n$ being able to generate another partition of $n$. We solve the problem of finding the minimum size partition for which the set of partitions this partition can generate contains all size-$k$ partitions of $n$. We describe how this result can be applied to solving a class of combinatorial optimization problems.
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a previous paper by the author , enable us to state and prove new generalizations of two of these results.
Wilf's Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as \emph{Wilf partitions}. Using $f(n)$ to denote the number of Wilf partitions, we establish lead-order asymptotics for $\ln{f(n)}$.
We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d, and the set of partitions where at least one part appears d or more times.
We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $μ$ one can force the strong polarized relation with respect to the pair $μ^+,μ$.
Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of Rødseth on enumerating minimal complete partitions.
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem …
The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition p…
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.
Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal ob…
Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant o…
We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.
We consider the problem of finding a minimum common partition of two strings (MCSP). The problem has its application in genome comparison. MCSP problem is proved to be NP-hard. In this paper, we develop an Integer Programming (IP) formulation for the problem and implement it. The experimental results are compared with the previous state-of-the-art algorithms and are found to be promising.
On the complexity of Sandwich Problems for $M$-partitions
本文研究了$M$-划分问题的复杂性，通过结构分类和高效算法，对矩阵划分和沙丁鱼问题进行了P与NP完全的分类。
Complexity Dichotomies for the Maximum Weighted Digraph Partition Problem
本文提出了最大加权有向图划分（MWDP）问题，并证明了其在任意、有向定向及对称有向图上的三个复杂性二分法，同时展示了该理论在二元行动polymatrix博弈及图论问题中的应用。
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