Electronic Notes in Discrete Mathematics最新文献

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On characterizing the extreme points of the generalized transitive tournament polytope 关于广义可传递竞赛多面体极值点的刻画

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.047

Konstantinos Papalamprou

A non-negative $n \times n$ matrix [ $x_{i j}$ ] is called generalized tournament, denoted GTT(n), if: $x_{i i} = 0$ (for all i), $x_{i j} + x_{j i} = 1$ (for all(i,j) with $i \neq j$ ) and $1 \leq x_{i j} + x_{j k} + x_{k i} \leq 2$ (for all (i,j,k) with $i, j, k$ pairwise distinct). In [9], using hypergraphs associated with GTT matrices, it has been shown that for $n \leq 6$ all the vertices of the GTT(n) polytope are half-integral. In this work, we show that these matrices belong to the class of 2-regular matrices and highlight the related optimization implications. Finally, based on our approach and known partial results, conjectures on characterizing the extreme points of the GTT(n) polytope for $n \geq 7$ are provided.

一个非负的n×n矩阵[xij]被称为广义竞赛，记作GTT(n)，如果:xii=0(对于所有i)， xij+xji=1(对于所有(i,j)且i≠j)， 1≤xij+xjk+xki≤2(对于所有(i,j,k)且i,j,k成对不同)。在[9]中，利用与GTT矩阵相关的超图，证明了当n≤6时，GTT(n)多面体的所有顶点都是半积分。在这项工作中，我们证明了这些矩阵属于2正则矩阵类，并强调了相关的优化含义。最后，根据我们的方法和已知的部分结果，给出了n≥7时GTT(n)多面体极值点表征的猜想。

引用次数: 0

Autoparatopism stabilized colouring games on rook's graphs 自斜视在rook的图形上稳定了着色游戏

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.040

Stephan Dominique Andres, Helena Bergold, Raúl M. Falcón

We introduce the autoparatopism variant of the autotopism stabilized colouring game on the $n \times n$ rook's graph as a natural generalization of the latter so that each board configuration is uniquely related to a partial Latin square of order n that respects a given autoparatopism (θ; π). To this end, we distinguish between $π \in {Id, (12)}$ and $π \in {(13), (23), (123), (132)}$ . The complexity of this variant is examined by means of the autoparatopism stabilized game chromatic number. Some illustrative examples and results are shown.

我们在n×n车图上引入自自论稳定着色游戏的自自自论变体，作为后者的自然推广，使得每个棋盘构型都与尊重给定自自自自性的n阶部分拉丁方唯一相关(θ;π)。为此,我们区分π∈{Id(12)}和π∈{(13)、(23)、(123)、(132)}。该变异的复杂性是通过自拟真稳定博弈色数来检验的。给出了一些说明性的例子和结果。

引用次数: 4

Two-line graphs of partial Latin rectangles 部分拉丁矩形的双线图

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.010

Eiran Danan, Raúl M. Falcón, Dani Kotlar, Trent G. Marbach, Rebecca J. Stones

Two-line graphs of a given partial Latin rectangle are introduced as vertex-and-edge-coloured bipartite graphs that give rise to new autotopism invariants. They reduce the complexity of any currently known method for computing autotopism groups of partial Latin rectangles.

给出了给定的部分拉丁矩形的双线图作为顶点和边缘的彩色二部图，并由此产生了新的自洽不变量。它们降低了任何目前已知的计算部分拉丁矩形自洽群的方法的复杂性。

引用次数: 6

The complete classification of empty lattice 4-simplices 空格单形的完全分类

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.027

Óscar Iglesias Valiño , Francisco Santos

In previous work we classified all empty 4-simplices of width at least three. We here classify those of width two. There are 2 two-parameter families that project to the second dilation of a unimodular triangle, $29 + 23$ one-parameter families of them that project to hollow 3-polytopes, and 2282 individual ones that do not.

在之前的工作中，我们分类了宽度至少为3的所有空4-简形。这里我们把宽度为2的分类。有2个双参数族投射到单模三角形的第二次膨胀，其中29+23个单参数族投射到中空的3-多面体，2282个单独的族没有。

引用次数: 0

Extended Lagrange's four-square theorem 扩展了拉格朗日四平方定理

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.036

J. Lacalle, L.N. Gatti

We prove the following extension of Lagrange's theorem: given a prime number p and $v_{1}, \dots, v_{k} \in Z^{4}, 1 \leq k \leq 3$ , such that ${∥ v_{i} ∥}^{2} = p$ for all $1 \leq i \leq k$ and $〈 v_{i} | v_{j} 〉 = 0$ for all $1 \leq i < j \leq k$ , then there exists $v = (x_{1}, x_{2}, x_{3}, x_{4}) \in Z^{4}$ such that $〈 v_{i} | v 〉 = 0$ for all $1 \leq i \leq k$ and $∥ v ∥ = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = p$ This means that, in $Z^{4}$ , any system of orthogonal vectors of norm p can be completed to a base. We conjecture that the result holds for every norm $p \geq 1$ .

我们证明了拉格朗日定理的以下推广:给定一个素数p和v1，…，vk∈Z4,1≤k≤3，使得∥vi∥2=p对于所有1≤i≤k，并且对于所有1≤i<j≤k， < vi|vj > =0，则存在v=(x1,x2,x3,x4)∈Z4使得对于所有1≤i≤k < vi|v > =0，并且∥v∥=x12+x22+x32+x42=p，这意味着在Z4中，任何范数p的正交向量系统都可以被补成一个基。我们推测，对于每个范数p≥1，结果都成立。

引用次数: 2

An improvement of the lower bound on the maximum number of halving lines in planar sets with 32 points 32点平面集合中最大减半线数下界的改进

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.052

Javier Rodrigo, Ma Dolores López

In this paper we give a recursive lower bound on the maximum number of halving lines for sets in the plane and as a consequence we improve the current best lower bound on the maximum number of halving lines for sets in the plane with 32 points.

本文给出了平面上集合的最大减半线数的一个递归下界，从而改进了目前平面上32点集合的最大减半线数的最佳下界。

引用次数: 1

Sums of finite subsets in Rd Rd中有限子集的和

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.012

Mario Huicochea

Let $B_{1}, B_{2}, \dots, B_{m}$ be nonempty finite subsets of $R^{d}$ with $B_{i}$ not contained in an affine hyperplane for each $i \in {2, 3, \dots, m}$ . First we get a sharp lower bound on $| B_{1} + B_{2} |$ when $| B_{2} | = d + 1$ . Using this result and other ideas, we find a nontrivial lower bound on $| B_{1} + B_{2} + \dots + B_{m} |$ which generalizes a result of M. Matolcsi and I. Z. Ruzsa [7].

设B1,B2，…，Bm是Rd的非空有限子集，且对于每个i∈{2,3，…，m}， Bi不包含在仿射超平面上。首先，当|B2|=d+1时，我们得到了一个明显的下界。利用这一结果和其他思想，我们得到了一个非平凡下界，推广了M. Matolcsi和I. Z. Ruzsa[7]的结果。

引用次数: 2

Iterated Sumsets and Olson's Generalization of the Erdős-Ginzburg-Ziv Theorem 迭代的集合和Olson对Erdős-Ginzburg-Ziv定理的推广

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.006

David J. Grynkiewicz

Let $G ≅ Z / m_{1} Z \times \dots \times Z / m_{r} Z$ be a finite abelian group with $m_{1} | \dots | m_{r} = \exp (G)$ . The Kemperman Structure Theorem characterizes all subsets $A, B \subseteq G$ satisfying $| A + B | < | A | + | B |$ and has been extended to cover the case when $| A + B | \leq | A | + | B |$ . Utilizing these results, we provide a precise structural description of all finite subsets $A \subseteq G$ with $| n A | \leq (| A | + 1) n - 3$ when $n \geq 3$ (also when G is infinite), in which case many of the pathological possibilities from the case $n = 2$ vanish, particularly for large $n \geq \exp (G) - 1$ . The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence S of terms from G having length $| S | \geq 2 | G | - 1$ must either have every element of G representable as a sum of $| G |$ -terms from S or else have all but $| G / H | - 2$ of its terms lying in a common H-coset for some $H \leq G$ . We show that the much weaker hypothesis $| S | \geq | G | + \exp (G)$ suffices to obtain a nearly identical conclusion, where for the case H is trivial we must allow all but $| G / H | - 1$ terms of S to be from the same H-coset. The bound on $| S |$ is improved for sever

设G = Z/ m1zx…×Z/mrZ是一个有限阿贝尔群，其中m1|…|mr=exp (G)。kempman结构定理描述了所有子集A、B的≥≥A+B; <|A|+|B|，并将其扩展到≤|A|+|B|的情况。利用这些结果，我们提供了当n≥3(也当G为无限大)时，所有具有|nA|≤(| a |+1)n−3的有限子集a的精确结构描述，在这种情况下，当n=2时的许多病态可能性消失，特别是当n≥exp (G)−1时。该结构描述与其他论证相结合，推广了Olson的子序列和结果，即由长度为|S|≥2|G|−1的G的项组成的序列S，要么G的每一个元素都可以表示为来自S的|G|项的和，要么除了|G/H|−2项以外的所有项都在H-余集内，对于某些H≤G。我们证明了弱得多的假设|S|≥|G|+exp (G)足以得到一个几乎相同的结论，其中对于H是平凡的情况，我们必须允许S的除|G/H|−1项以外的所有项都来自同一个H集。改进了若干类群G上的界，得到了最优下界。

引用次数: 0

Structure and Enumeration of K4-minor-free links and link diagrams k4 -非次要链接和链接图的结构和枚举

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.021

Juanjo Rué , Dimitrios M. Thilikos , Vasiliki Velona

We study the class $L$ of link types that admit a K₄-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K₄. We prove that $L$ is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate $L$ and subclasses of it, with respect to the minimal number of crossings or edges in a projection of $L \in L$ . Further, we enumerate (both exactly and asymptotically) all connected K₄-minor-free link diagrams, all minimal connected K₄-minor-free link diagrams, and all K₄-minor-free diagrams of the unknot.

我们研究了一类允许有K4-次要无图的链路类型，即它们可以被投影到平面上，使得得到的图不包含K4的任何细分。证明了L是环面连杆的一个子类在连通和作用下的闭包。利用这一结构结果，我们列举了L及其子类，关于L∈L的投影中交叉或边的最小数量。进一步，我们(精确地和渐近地)列举了所有连通的k4 -小自由连杆图，所有最小连通的k4 -小自由连杆图，以及所有解结的k4 -小自由连杆图。

引用次数: 1

Kirchhoff index of the connections of two networks by an edge 两个网络通过一条边连接的基尔霍夫指数

Q2 Mathematics

Electronic Notes in Discrete Mathematics

Pub Date : 2018-07-01 DOI: 10.1016/j.endm.2018.06.049

Silvia Gago

In this work we compute the group inverse of the Laplacian of the connections of two networks by and edge in terms of the Laplacians of the original networks. Thus the effective resistances and Kirchhoff index of the new network can be derived from the Kirchhoff indexes of the original networks.

在这项工作中，我们根据原始网络的拉普拉斯算子计算了两个网络by和edge连接的拉普拉斯算子的群逆。因此，新网络的有效阻力和Kirchhoff指数可以由原网络的Kirchhoff指数推导出来。

引用次数: 2

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