The crossing number of cubes with small order

Cubicity, Degeneracy, and Crossing Number

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the interse…

Number cubes with consecutive line sums

We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.

On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq C…

An analogue of Gromov's waist theorem for coloring the cube

It is proved that if we partition a $d$-dimensional cube into $n^d$ small cubes and color the small cubes into $m+1$ colors then there exists a monochromatic connected component consisting of at least $f(d, m) n^{d-m}$ small cubes.

多项式恒等性特征化daisy立方体的问题

Resolving problems on polynomial characterizations of daisy cubes and extensions

本文研究daisy立方体的多项式特征，证明了在特定条件下，daisy立方体与顶点0^n等价，并解决了两个开放问题。

The number of small covers over cubes

In the present paper we find a bijection between the set of small covers over an $n$-cube and the set of acyclic digraphs with $n$ labeled nodes. Using this, we give a formula of the number of small covers over an $n$-cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and $\mathbf{Z}^n$-equivariant diffeomorphism classes. Moreover we prove that the number of acyclic digraphs with $n$ unlabeled nodes is an upper bound of the number of small covers over an $n$-cube up to diffeomorphism.

On the Crossing Numbers of Cartesian Products of Small Graphs with Paths, Cycles and Stars

There has been significant research dedicated towards computing the crossing numbers of families of graphs resulting from the Cartesian products of small graphs with arbitrarily large paths, cycles and stars. For graphs with four or fewer vertices, these have all been computed, but there are still various gaps for graphs with five or more vertices. We contribute to this field by determining the crossing numbers for fifteen such families.

On Genus g Orientable Crossing Numbers of Small Complete Graphs

The current state of knowledge of genus g orientable crossing numbers of complete graphs through K11 is reviewed. It is shown that cr3(K10)=3, cr3(K11)<=14, and cr4(K11)=4. It is established with the aid of an algorithm that there are precisely two non-isomorphic embeddings of K9 with a hexagon with all its vertices distinct on a surface of genus 3.

在广义汉明立方体上的bootstrap渗流

Bootstrap percolation on a generalized Hamming cube

研究了在具有顶点集V={0,1}^n且边连接汉明距离为1,2,…,k的图上的r-邻居bootstrap渗流过程，确定了r=2,3时渗流临界概率的渐进行为，并在确定性设定下获得了k≥2时最小渗流集大小的多个结果。

Asymptotically efficient triangulations of the d-cube

Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times Δ^m$, for a (small) natural number $m$ of our choice. $Δ^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assum…

Half-arc-transitive graphs of prime-cube order of small valencies

A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$…

The crossing number of the cone of a graph

Motivated by a problem asked by Richter and by the long standing Harary-Hill conjecture, we study the relation between the crossing number of a graph $G$ and the crossing number of its cone $CG$, the graph obtained from $G$ by adding a new vertex adjacent to all the vertices in $G$. Simple examples show that the difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In this work, we are interested in finding the smallest possible difference, that is, for each non-negative integer $k$, find the smallest $f(k)$ for which there exists a graph with crossing number at least $k$…

完全图的书交叉数与小局部凸交叉数

Book crossing numbers of the complete graph and small local convex crossing numbers

本研究改进了完全图$K_n$在$k\neq 14$时的$k$页书交叉数下界，确定了$2 < n/k \neq 3$时的书交叉数，并给出局部交叉数$\neq 4$的凸图最大边数。

On the Ramsey number of the triangle and the cube

The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K_3,Q_n) = 2^{n+1} - 1 for every n \in \N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K_3,Q_n) \le 7000 \cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n \to \infty.

Note on k-planar crossing numbers

The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to th…

Coloring the cube with rainbow cycles

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that $$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$ Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod…

Corners and simpliciality in oriented matroids and partial cubes

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of un…

On the Crossing Number of the Cartesian Product of a Sunlet Graph and a Star Graph

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given.

限定可折叠成立方体的矩形多联骨牌所需的孔洞数量

Bounding the number of holes required for folding rectangular polyominoes into cubes

该研究针对带孔矩形多联骨牌沿网格线折痕能否折叠成立方体的问题，探究保证可折叠性的最小孔洞集合，发现同型孔洞时最小集合规模至多为4，异型孔洞时规模无上限。

Monochromatic Hilbert cubes and arithmetic progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geq 3$ be an integer. Then for every $ε>0$, …

Octopuses in the Boolean cube: families with pairwise small intersections, part I

Let $\mathcal F_1, \ldots, \mathcal F_\ell$ be families of subsets of $\{1, \ldots, n\}$. Suppose that for distinct $k, k'$ and arbitrary $F_1 \in \mathcal F_{k}, F_2 \in \mathcal F_{k'}$ we have $|F_1 \cap F_2|\le m.$ What is the maximal value of $|\mathcal F_1|\ldots |\mathcal F_\ell|$? In this work we find the asymptotic of this product as $n$ tends to infinity for constant $\ell$ and~$m$. This question is related to a conjecture of Bohn et al. that arose in the 2-level polytope theory and asked for the largest product of the number of facets and vertices in a two-level polytope. This conje…

There are only two nonobtuse binary triangulations of the unit $n$-cube

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit c…

Computational search of small point sets with small rectilinear crossing number

Let $\crs(K_n)$ be the minimum number of crossings over all rectilinear drawings of the complete graph on $n$ vertices on the plane. In this paper we prove that $\crs(K_n) < 0.380473\binom{n}{4}+Θ(n^3)$; improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of $K_n$ for small values of $n$, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The "small" sets where found using a simple heuristic detailed in this paper.

具有两个小划分类的图的三部圆交叉数

The tripartite-circle crossing number of graphs with two small partition classes

本文研究三部图在三部圆绘制下的交叉数问题，并给出了当两个划分类大小不超过2时完全三部图 $K_{a,b,n}$ 的三部圆交叉数精确值。

Higher dimensional digraphs from cube complexes and their spectral theory

We define $k$-dimensional digraphs and initiate a study of their spectral theory. The $k$-dimensional digraphs can be viewed as generating graphs for small categories called $k$-graphs. Guided by geometric insight, we obtain several new series of $k$-graphs using cube complexes covered by Cartesian products of trees, for $k \geq 2$. These $k$-graphs can not be presented as virtual products, and constitute novel models of such small categories. The constructions yield rank-$k$ Cuntz-Krieger algebras for all $k\geq 2$. We introduce Ramanujan $k$-graphs satisfying optimal spectral gap property, a…

Random subgraphs of finite graphs: III. The phase transition for the $n$-cube

We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λ2^{n/3}$, where $λ$ is a small positive constant. Let $ε=n(p-p_c(n))$. In two previous papers, we showed that the largest cluster inside a scaling window given by $|ε|=Θ(2^{-n/3})$ is of size $Θ(2^{2n/3})$, below this scaling window it is at most $2(\log2) nε^{-2}$, and above this scaling window it is at most $O(ε2^n)$. In this paper, we prove that for $p - p_c(n) \geq e^…

The chromatic number of the square of the 8-cube

A cube-like graph is a Cayley graph for the elementary abelian group of order $2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This coloring problem can be considered in the framework of coding theory, as the graph $Q_n^k$ can be constructed with one vertex for each binary word of length $n$ and edges between vertices exactly when the Hamming distance between the corresponding words is at most $k$. Consequently, a proper coloring of $Q_n^k$ corresponds to a partition of the $n$-dimensional binary…

On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar.

模形式的收缩圈在交叉数上达到极值

Modular systoles are extremal for the crossing number

本文研究同余曲面上收缩圈的交叉数，证明其相交增长速率在同拓扑曲面且相同基数的所有曲线集合中达到最优最小。

魔方群直径的图论估计

Graph-theoretical estimates of the diameters of the Rubik's Cube groups

本文提出一种基于图阶数、度数、围长及过顶点圈数等局部参数的对称图直径严格下界方法，并应用于魔方群凯莱图，获得比随机正则图下界更紧、达到正确直径60%至77%的估计。