S. Foldes, S. Radeleczki
Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in a linearly ordered set. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions of a set, in a general sense of intervals defined axiomatically. This characterization is used to show that the lattice of interval decompositions is balanced.
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We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous grammars that allow to generate tree-like structures and obtain fixed-point functional equations to enumerate these. We also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the Tamari lattice. Finally, we investigate other families of binary trees that are also close…
We find the first non-octahedral balanced 2-neighborly 3-sphere and the balanced 2-neighborly triangulation of the lens space $L(3,1)$. Each construction has 16 vertices. We show that there exists a balanced 3-neighborly non-spherical 5-manifold with 18 vertices. We also show that the rank-selected subcomplexes of a balanced simplicial sphere do not necessarily have an ear decomposition.
We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.
We give a new proof of König's theorem and generalize the Gallai-Edmonds decomposition to balanced hypergraphs in two different ways. Based on our decompositions we give two new characterizations of balanced hypergraphs and show some properties of matchings and vertex cover in balanced hypergraphs.
A decomposition theorem for balanced measures
本文研究连通图上平衡概率测度的结构，证明其可分解为若干“基本”极端平衡测度的凸组合，并给出基本平衡测度数量的本质紧上界。
Given an $n\times n$ array $M$ ($n\ge 7$), where each cell is colored in one of two colors, we give a necessary and sufficient condition for the existence of a partition of $M$ into $n$ diagonals, each containing at least one cell of each color. As a consequence, it follows that if each color appears in at least $2n-1$ cells, then such a partition exists. The proof uses results on completion of partial Latin squares.
Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipar…
A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced $k$-partite graph if $|V_{1}| = |V_{2}| =... = |V_{k}|$. An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorabl…
In a balanced graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local balance conditions in graph decompositions. In the basic case where a single graph with colored loops is used as a block, an existence theory for such decompositions follows as a straightforward generalization of previous work on balanced graph decompositions.
A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-…
A {\em balanced coloring} of a graph $G$ means a triple $\{P_1,P_2,X\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \uplus P_2 \uplus X$ and $|P_1|=|P_2|$. A {\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \uplus P_2 \uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \uplus \cdots \uplus V_r$ (for some $r$) such that, for every $i \in \{1,\cdots,r\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \cap P_1| = |V_i \cap P_2|$. Then the {\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ suc…
We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved kernelization and approximation al…
Fractional clique decompositions of dense balanced multipartite graphs
研究密集平衡多部图的分数 clique 分解条件，提出新的度数阈值，适用于多于 s 部的多部图。
Coarse Balanced Separators and Tree-Decompositions
本文针对 $K_{t,t}$ 诱导子图无关图，在分离器由有界数量半径为1的球组成时，证实了关于粗平衡分离器与有界树宽图拟等距的猜想，从而沟通了两个关于排除平面图作为诱导子式的图结构猜想。
Balanced rectangles over Sturmian words and minimal discrepancy intervals
研究Sturmian单词斜率为α的m×n矩形矩阵的平衡性质，通过Ostrowski表示描述其特性，扩展了Anselmo等人和Shallit的研究，采用nα mod 1的分布方法。
An antichain of subsets is a set of subsets such that no subset in the antichain is a proper subset of any other subset in the antichain. The Dedekind number counts the total number of antichains of subsets of an n-element set. This paper investigates the interval structure of the lattice of antichains. Several partitioning theorems and counting formulas for the size of intervals are derived.
An $α$-greedy balanced pair in an ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of greedy linear extensions of $P$ that put $x$ before $y$ among all greedy linear extensions is in the real interval $[α, 1-α]$. We prove that every $N$-free ordered set which is not totally ordered has a $\frac{1}{2}$-greedy balanced pair.
$f$-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions
本文证明了均衡单纯复形的$f$-向量可展现类几何Lefschetz分解的布尔分解，基于Nevo--Petersen猜想将其与旗球面多项式的正定性及几何问题相联系，并指出分解次数未减半。
Degree-balanced decompositions of cubic graphs
本文证明了任意三次图存在一个生成子图，使各度数顶点数接近 n/4（偏差至多 1/2，最多三个例外），从而解决了 Alon 和 Wei 的猜想。
Balanced Stick Breaking
针对单位圆上点序列折分区间的长度比问题，本文通过van der Corput序列的精细化偏差估计，证明r个连续区间的长度比可达$1+c\log{r}/r$，证实Brethouwer猜想。
On the Existence of Balanced Chain Rule Task Sets
本文使用图分解方法，扩展完全有向图分解结果，为微积分I构建了评估链式法则的平衡混合练习习题集。
The interval posets of permutations seen from the decomposition tree perspective
本文从排列分解树视角研究区间偏序集，给出从分解树构造该偏序集的方法，并用符号与解析组合学解决相关开放计数问题，同时计算其 Möbius 函数。
A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two degree groups are placed as uniformly as possible. We prove that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the num…
Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps
通过建立贪婪Tamari区间与二部平面地图的同构递归分解，给出了贪婪1-Tamari区间与平面2-星座等计数的组合证明，并提出了更一般的猜想。
Triangle decompositions of PG(n-1,2)
本文在射影空间 PG(n-1,2) 中构造平衡三角形设计，并证明 Singer 循环群不变设计与特殊 18 元子集划分等价，为 n=7,13,19 给出实例。
Delannoy--Steinhaus triangles over $\mathbb{Z}/2\mathbb{Z}$: weight spectrum, balanced triangles, and extremal values
本文研究 $\boldsymbol{\text{Z}/2\boldsymbol{\text{Z}}}$ 上的 Delannoy–Steinhaus 三角形，推导其权重公式，确定完整权重谱，刻画平衡三角形存在条件，并计算各类极值权重。
On coarse tree decompositions and coarse balanced separators
研究将图的树宽与平衡分隔器数的线性依赖推广到粗图论场景，证明存在粗平衡分隔器的图有对应粗树分解，且加倍维度受限图中粗平衡分隔器与粗树分解等条件等价。
In this paper, we study the decomposition of Bruhat intervals in a Coxeter group with respect to cosets of a parabolic subgroup. Our main result is that the intersection of a lower Bruhat interval with a parabolic coset contains a unique maximal element. As an application, we give a decomposition formula for the Poincaré polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.
A balanced pair in a finite ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We prove that every finite $N$-free ordered set which is not totally ordered has a balanced pair.
Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of Möbius decomposition space, a far-reaching generalisation of the notion of Möbius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but is not induced by a Möbius category), can be realised as…
The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals
本文证明了BBDVW猜想在下区间中的成立，为Kazhdan-Lusztig多项式组合不变猜想提供了新的证据。
Let $P$ be a set of points in $\mathbb{R}^d$, $B$ a bicoloring of $P$ and $\Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An object from $\Oo$ is called balanced with respect to $B$ if it contains the same number of points from each color of $B$. For a collection $\B$ of bicolorings of $P$, a geometric system of unbiased representatives (G-SUR) is a subset $\Oo'\subseteq\Oo$ such that for any bicoloring $B$ of $\B$ there is an object in $\Oo'$ that is balanced with respect to $B$. We study the problem of finding G-SURs. We obtain general bounds on the size of G-SUR…
Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns
本文研究并枚举了满足特定钩长可除条件的划分，证明了晶体反射保持d-平衡性质，并给出了e-正则划分在块中的计数公式。
Even cycle decomposition thresholds for dense multipartite graphs
本文确定了稠密2-平衡多部图中偶环分解的阈值：对任意r≥2，C4分解阈值为2/3，C_{2k}（k≥3）分解阈值为1/2。
A generalised block decomposition theorem
本文研究了全序集乘积偏序集的一类线性表示，证明其可分解为块区间的区间表示，并通过“中间正合性”这一同调性质进行刻画。
We define two recursive functions obtained by decomposition of a given interval into four close parts and prove two lemmas which determine features of these functions.
We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k colors so that every translate of the negative octant containing at least k^6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition pr…
Extremal decompositions of tropical varieties and relations with rigidity theory
针对有理平衡权重热带簇的极值分解计算难题，本文提出快速算法，并建立其与刚性理论的联系，证明热带超曲面极值等价于其倒数图在位似下唯一。
A characterization of interval nest digraphs
本文通过嵌套序列为特征，完整刻画了区间嵌套有向图的结构特性。
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces
本文证明了 Gálvez-Kock-Tonks 猜想在局部离散分解空间上的情形，通过构造局部严格模型并利用 2-范畴论证，表明 Lawvere 区间构造仅允许恒等自修改，为一般猜想提供了首个实质证据。
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