关于折叠超立方体中完美匹配与子超立方体之间的双射猜想

Star Structure Connectivity of Folded hypercubes and Augmented cubes

The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $κ(G; H)$ (resp. $H$-substructure connectivity $κ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ…

折叠超立方体的分支边连通度

Component edge connectivity of the folded hypercube

不完全图$G$的$g$ -分量边连通度$cλ_g(G)$是指删除其导致图至少具有$g$分量的边的最小数目.本文确定了折叠超立方体$g\leq 2^{[\frac{n+1}2]}$的分支边连通度，其中$g$为正整数，$g=\sum\limits_{i=0}^{s}2^{t_i}$为$g$的分解，使得$t_0=[\log_{2}{g}],$和$t_i=[\log_2({g-\sum\limits_{r=0}^{i-1}2^{t_r}})]$为$i\geq 1$ .

超立方体双射的尖锐内积相关性

Sharp Inner Product Correlations for Hypercube Bijections

研究超立方体双射的内积相关性，证明了双射下非负内积概率的下界，通过谱分解和线性规划方法分析。

The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes

As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $κ(G, T)$ (resp. $T$-substructure connectivity $κ^{s}(G, T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. [6] showed $κ(Q_{n},K_{1,1})=κ^{s}(Q_{n},K_{1,1})=n-1$ and $κ(Q_{n},K_{1,r})=κ^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for $2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that $κ(Q_{n},K_{1,4})=κ^{…

边序图中的增路：超立方体和随机图

Increasing paths in edge-ordered graphs: the hypercube and random graphs

图$G=(V,E)$的边序是双射$ϕ:E\to\{1,2,...,|E|\}$。给定一个边序，一个边序列$P=e_1,e_2,...,e_k$是一条增路，如果它是$G$中的一条路，对所有$i<j$满足$ϕ(e_i)<ϕ(e_j)$。对于图$G$，设$f(G)$是最大整数$\ell$，使得$G$的每个边序都包含一条长度为$\ell$的递增路。参数$f(G)$首先研究了$G=K_n$，随后研究了其他家庭的图。本文给出了超立方体和随机图$G(n,p)$的$f$的界.

On Mappings on the Hypercube with Small Average Stretch

Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$, and let $ϕ\colon \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $ϕ$ as ${\sf avgStretch}(ϕ) = {\mathbb E}[{\sf dist}(ϕ(x),ϕ(x'))]$, where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate. In this paper we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results. (1) For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $ϕ_A \colon \{0,1\}^{n-1} \to A$ such t…

超立方体矩形划分的枚举与渐进行为公式

Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

本文研究了超立方体划分的枚举问题，通过正交划分和满$p$-ary树的双射，推导了卡特兰数的广义渐进行为公式。

图的最大反强迫数的紧上界

Tight upper bound on the maximum anti-forcing numbers of graphs

本文给出图的最大反强迫数的新上界，刻画极值图及其好完美匹配，并证明极值图可由K2经两种扩张构造，且完全图、完全二部图及三类超立方体的好完美匹配数量被精确确定。