Zhiyi Tang
We study the problem of Embedding Wirelength of $n$-dimensional Hypercube $Q_n$ into Cylinder $C_{2^{n_1}}\times P_{2^{n_2}}$, where $n_1+ n_2=n$, called EWHC. We show that such wirelength corresponding to Gray code embedding is $2^{n_2}(3\cdot 2^{2n_1-3}-2^{n_1-1})+2^{n_1} (2^{2n_2-1}-2^{n_2-1})$. In addition, we prove that Gray code embedding is an optimal strategy of EWHC.
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A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2. Basic necessary conditions on Hamiltonian embeddings for $Q_n$ and conjectures are made about other values of $n$.
Though embedding problems have been considered for several regular graphs, it is still an open problem for hypercube into torus. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.
Hypercube emulation of interconnection networks topologies
我们解决各种拓扑结构（德Bruijn，弦环，广义彼得森，网格）在超立方体或半超立方体的各种方式（等距嵌入，嵌入规模，嵌入距离）。获得的嵌入示例：超立方体可嵌入冒泡排序和双弦环拓扑以及规则映射的无穷级数。
For a set $X$ of binary words of length $h$ the daisy cube $Q_h(X)$ is defined as the subgraph of the hypercube $Q_h$ induced by the set of all vertices on shortest paths that connect vertices of $X$ with the vertex $0 ^h$. A vertex in the intersection of all of these paths is a minimal vertex of a daisy cube. A graph $G$ isomorphic to a daisy cube admits several isometric embeddings into a hypercube. We show that an isometric embedding is proper if and only if the label $0 ^h$ is assigned to a minimal vertex of $G$. This result allows us to devise an algorithm which finds a proper embedding o…
Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed i…
A long-standing conjecture on spanning trees of a hypercube states that a balanced tree on $2^n$ vertices with maximum degree at most $3$ spans the hypercube of dimension $n$ \cite{havel1986}. In this paper, we settle the conjecture for a special family of binary trees. A $0$-caterpillar is a path. For $k\geq 1$, a $k$-caterpillar is a binary tree consisting of a path with $j$-caterpillars $(0\leq j\leq k-1)$ emanating from some of the vertices on the path. A $k$-caterpillar that contains a perfect matching is said to be perfectly balanced. In this paper, we show that a perfectly balanced $2$-…
The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-α$, where $α=0$ if vertices $u$ and $v$ form a normal vertex-pair and $α=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).
Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)
研究判别排列交叠格L(B(n,k))的度量结构，证明其覆盖图可嵌入超立方体，成为部分立方体和中位图，揭示超立方体几何。
Optimal Embeddings of Posets in Hypercubes
定义偏序集P的超立方体高度h*和宽度w*，证明w*≤|P|的猜想，使用Hall定理修改已有嵌入。
Embedding multidimensional grids into optimal hypercubes
令$G$和$H$是图，其中$|V(H)|\geq |V(G)| $和$f:V(G)\rightarrow V(H)$是它们的顶点的一对一映射。设$dilation(f) = max\{ dist_{H}(f(x),f(y)): xy\in E(G) \}$，其中$dist_{H}(v,w)$是$H$的顶点$v$和$w$之间的距离。现在令$B(G,H)$ = $min_{f}\{ dilation(f) \}$，在所有这样的映射$f$上。参数$B(G,H)$是$G$的经典和充分研究的“带宽”的推广，定义为$B(G,P(n))$，其中$P(n)$是$n$点和$n = |V(G)|$上的路径。令$[a_{1}\times a_{2}\times \cdots \times a_{k} ]$是$k$维网格图，在第$i$维坐标中具有整数值$1$到$a_{i}$.本文研究了当$G = [a_{1}\times a_{2}\times \cdots \times a_{k} ]$和$H$是维数为$n = \lceil log_{2}(|V(G)|) \rceil$的超立方体$Q_{n}$时的$B(G,H)$，其中最小维数的超立方体的点数至少与$G$一样多.我们的主要结果是，$$B( [a_{1}\times a_{2}\times \cdots \times a_{k} ],Q_{n}) \le…
We study embeddings of the $n$-dimensional hypercube into the circuit with $2^n$ vertices. We prove that the circular wirelength attains minimum by gray coding, which is called the CT conjecture by Chavez and Trapp (Discrete Applied Mathematics, 1998). This problem had claimed to be settled by Ching-Jung Guu in her doctor dissertation "The circular wirelength problem for hypercubes" (University of California, Riverside, 1997). Many people argue there are gaps in her proof. We eliminate gaps in her dissertation.
Hypercube embedding of Wythoffians
Wythoff构造采用一个$d$维多面体$P$（$\{0,..., d\}$的子集$S$），并返回另一个$d$维多面体$P(S)$ .如果$P$是正则多面体，则$P(S)$是顶点传递的。这种构造在3维和4维中构建了阿基米德多面体和平铺的大部分。我们想确定，那些Wythoffians $P(S)$与常规$P$有他们的骨架或对偶骨架等距嵌入到超立方体$H_m$和半立方体${1/2}H_m$。我们发现六个无穷级数，我们推测，涵盖了所有的情况下，尺寸$d>5$和一些零星的情况下，在3和4（见表\ref{WythoffEmbeddable3}和\ref{WythoffEmbeddable4}）。这六个无穷级数中的三个可以用Wythoff构造嵌入Coxeter群的一般结果来解释。在最后一节中，我们考虑了欧几里德的情况下，也可嵌入$P(S)$的zonotopality解决整个文本。
We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects functions that are $\varepsilon$-far from monotone with probability at least $2/3$. 2. For all $m\in\mathbb{N}$, we show an $\tilde{O}(\sqrt{n}m^3/\varepsilon^2)$ query monotonicity tester over $[m]^n$. We also establish corresponding directed isoper…
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
本文利用增强立方体的凯莱性质和互反完美匹配方法，给出AQ_n中Q_n同构子图数量的下界及最少公共边子图对的构造，并应用于证明AQ_n的边不相交哈密顿圈分解和容错双泛圈性。
Vertex Turán problems in the hypercube
令$\mathcal{Q}_n$是$n$维超立方体：顶点集为$\{0,1\}^n$且顶点之间的边恰好在一个坐标上不同的图。对于$1\leq d\leq n$和$F\subseteq \{0,1\}^d$，如果每个嵌入$i:\{0,1\}^d\to \{0,1\}^n$满足$i(F)\not\subseteq S$，则我们说$S\subseteq \{0,1\}^n$是\emph{$F$-free}。我们考虑的问题有多大的$S\subseteq \{0,1\}^n$可以，如果它是$F$免费的。特别地，我们推广了这一领域的主要先验结果，对于$F=\{0,1\}^2$，由于E.A. Kostochka，并证明了近极值集结构的局部稳定性结果。我们还表明，所需的密度，以保证嵌入式副本的至少一个家庭的禁止配置可能会显着低于所需的，以确保嵌入式副本的家庭的任何个别成员。最后，我们证明了如果$n$足够大，正密度的$n$维超立方体的任何子集都将包含来自某个嵌入的$n$维子立方体的指数级多个点.
Large dilates of hypercube graphs in the plane
该研究针对平面中与n维单位超立方体1-骨架同构的距离图$Γ_n$，证明正上Banach密度可测集包含其所有足够大的伸缩图，首次给出树外可在正密度集实现维度最优嵌入的距离图实例。
Isometric embeddings of resonance graphs as finite distributive lattices
该研究针对平面二分图的共振图等距嵌入问题，证明了连通共振图的等距维数下界，给出等号成立的刻画，并设计算法生成对应二进制编码实现等距嵌入，解答了相关公开问题的基本情形。
Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes $Q_n$. They are induced with intervals between chosen vertices of $Q_n$ and the vertex $0^n\in V(Q_n)$. In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klavžar and Mollard, 2018, in the introductory paper of daisy cubes \cite{KlaMol-18}. To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph $G$ isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we pre…
Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the wirelength. In this manuscript, we obtain the wirelength (exact and minimum) of embedding complete multi-partite graphs into Cartesian product of paths and cycles, which include $n$-cube, $n$-dimensional mesh (grid), $n$-dimensional cylinder and $n$-dimensional torus, etc., as the subfamilies.
Binary Stretch Embedding of Weighted Graphs
本文针对加权图提出二进制拉伸嵌入问题，利用Hadamard码与线性规划给出上下界及精确解，并应用于改进Lee度量码的最大规模。
The Sign Pattern Problem for Ehrhart Polynomials
本文研究了Ehrhart多项式系数的符号模式问题，证明了在特定维度下存在多面体使低次系数为负而高次系数为正，并解决了7、8、9维度的符号模式问题。
Congestion bounds via Laplacian eigenvalues and their application to tensor networks with arbitrary geometry
本文研究了将任意图顶点嵌入树中的拥堵问题，利用图的拉普拉斯特征值给出拥堵界限，并应用于张量网络，对超立方体、晶格图等进行了精确分析，提出高效的收缩方法。
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