p-不连通图的重构

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ e…

不连通图的群不规则强度

Group irregularity strength of disconnected graphs

本文研究了图的群不规则强度，定义了使任意Abel群存在边标签函数令顶点边标签和互异的最小值，并给出了无小星连通分支图的界限与精确值结果。

Asymptotic Density of Graphs Excluding Disconnected Minors

For a graph $H$, let $$c_{\infty}(H)= \lim_{n \to \infty}\max\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\infty}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\infty}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\infty}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint copies of th…

通过超饱和度研究不连通公共图

Disconnected Common Graphs via Supersaturation

本文通过超饱和度方法研究不连通图的公共性，发现多个K₃和K₂的不相交并集可能为公共图，并改进了相关Ramsey多重性常数的界。

The extremal function for disconnected minors

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.

Antimagic orientations of disconnected even regular graphs

A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\ldots,m\}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u \in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$\ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been …

Factors of disconnected graphs and polynomials with nonnegative integer coefficients

We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has $n$ connected components, where $n$ is prime, or $n=1,4,8,9$, and satisfies some additional conditions, it factors uniquely under the given products. If, on the contrary, $n=6$ or 10, all cases of nonunique factorisation are described precisely.

非连通图的无标号列表染色函数

The unlabeled list color function of disconnected graphs

研究无标号图的列表染色函数，针对Kaul和Mudrock提出的在无标号情况下Donner结果的类似情况是否成立的问题，证明了\(n\)个顶点无边图的相关猜想，还表明若连通分量满足，则非连通图也满足该结果的无标号类似情况。

4-连通平面图中的不连通割集

Disconnected cuts in 4-connected planar graphs

本文研究4-连通平面图的极小割集，证明其所有极小割集连通当且仅当该图为近三角剖分，并基于此给出寻找不连通割集的线性时间算法框架。

A zonotope and a product of two simplices with disconnected flip graphs

We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of $Δ^4 \times Δ^n$ is not connected by flips for large $n$. Our proof makes use of a non-explicit probabilistic construction.

Disconnected Cuts in Claw-free Graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms are known for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to t…

Disconnected Character graphs and odd Dominating sets

Suppose $Γ$ is a finite simple graph. If $D$ is a dominating set of $Γ$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $Γ$, then we say that $D$ is an odd dominating set for $Γ$. For a finite group $G$, let $Δ(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that the complement of $Δ(G)$ contains an odd dominating set, if and only if $Δ(G)$ is a disconnected graph with non-bipartite complement.

Recolouring planar graphs of girth at least five

For a positive integer $k$, the $k$-recolouring graph of a graph $G$ has as vertex set all proper $k$-colourings of $G$ with two $k$-colourings being adjacent if they differ by the colour of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the $7$-recolouring graphs of planar graphs, the $5$-recolouring graphs of triangle-free planar graphs and the $4$-recolouring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose $6$-recolouring graph is disconnected, triangle-free planar graphs whose $4$-re…

不连通图与可实现距离序的极值界

Disconnected graphs and extremal bounds for realizable distance orders

本文研究不连通图的边序可实现性，发现K_n⊔K_n存在边序在ℝ^{n-2}不可实现但在ℝ^{n-1}可实现，还刻画了两环不交并的可实现性，给出对应最大边数的渐近界。

错排图中的不连通团

Disconnected Cliques in Derangement Graphs

本文建立了正交拉丁方与错排图中不连通极大团之间的对应关系，利用谱聚类方法得到模条件，并以此简洁证明了欧拉36军官问题无解。

关于满足Pálfy不等式的 disconnected character degree graphs 的数量

On the Number of Disconnected Character Degree Graphs Satisfying Pálfy's Inequality

本文研究了具有 disconnected character degree graph 的有限可解群，计算了满足 Pálfy 不等式的连通分支大小对的数量，并展示了对于固定正整数 $n$，恰好有 $n$ 个大小对满足不等式的不同图的阶数。

覆盖不连通多重图的计算复杂度

Computational Complexity of Covering Disconnected Multigraphs

本文针对不连通图覆盖定义推广后不等价的问题，比较三种不连通图覆盖定义的判定复杂度，证明满射覆盖与均匀覆盖满足按连通组件继承复杂度的性质，并指出均匀覆盖更自然、二者固定参数复杂度不同。

复曲面Hilbert格式的组合数学

Combinatorics of the toric Hilbert scheme

复曲面希尔伯特格式是一个参数空间，它适用于所有具有相同多阶希尔伯特函数的理想。与经典的Hilbert格式不同，环面Hilbert格式是否连通是未知的。我们构造了一个图上的所有单项理想的计划，称为翻转图，并证明了复曲面希尔伯特计划是连通的当且仅当翻转图是连通的。这些图用来表示P^4中的曲线，其相关的复曲面希尔伯特格式具有任意维数。我们表明，翻转图映射到鲍斯图的点配置定义的复曲面理想的所有三角剖分。受最近发现的一个断开的Baues图的启发，我们关闭的结果表明存在一个断开翻转图，因此一个断开复曲面希尔伯特计划。

关于离散 Morse 理论技术在不连通图复形上的一个应用的注记

A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs

本文指出 Forman 在不连通图复形上构造离散梯度向量场时所用验证断言存在问题，给出反例，并用更具一般性的观察重新证明该向量场确为梯度场。

Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing eq…

以完全不连通图作为星补的极大符号二部图

Maximal signed bipartite graphs with totally disconnected graphs as star complements

本文研究以完全不连通图为星补的符号二部图的极大阶，利用Hadamard和Conference矩阵给出存在条件与极大阶公式，并刻画极图。

On Andreae's Ubiquity Conjecture

A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

星图与路径的直积是反魔法的

The direct product of a star and a path is antimagic

本文研究星图与路径直积所得不连通图的反魔法标号问题，证明该类图具有反魔法性，为不连通图的反魔法性研究提供了新结果。

全不连通局部紧群中的几何顺从性

Geometric amenability in totally disconnected locally compact groups

本文给出拟传递图顺从性等价于其自同构群顺从且幺模的几何证明，强化了紧生成全不连通局部紧群的相关结论，引入几何顺从性与一致几何非顺从性概念并研究其与群作用、模同态的关联。

独立数为5的图的结构分类

Structural Classification of a Graph with Independence Number Five

本文针对独立数为5的图，利用独立多项式在图字典积下的封闭性及不连通图的独立多项式分解性质，对其独立多项式进行分类并刻画不连通构型。

高围长图与无H图中的匹配割

Matching Cuts in Graphs of High Girth and H-Free Graphs

该研究解决了匹配割问题的20年公开难题，证明多个匹配割相关问题在高围长有界度二部图、特定无H图上是NP完全的，还得到完美匹配割在无H子图图上的完整复杂度分类。

随机图中给定度序列的连通概率是多少？

What is The Probability That A Random Graph With A Given Degree Sequence is Connected?

研究随机图在给定度序列下连通的概率，通过分析度数为1或2的顶点数量，得出当低度顶点数量较少时图连通的概率较高，否则概率较低。

Björner和Lovász关于$χ$ -着色复形猜想的一个反例

A counterexample to a conjecture of Björner and Lovász on the $χ$-coloring complex

与色数为$χ$的每个图$G$相关联的是另一个图$G'$。$G'$的顶点集由$G$的所有$χ$ -着色组成，并且当两个$χ$ -着色恰好在一个顶点上不同时，它们是相邻的。根据Björner和Lovász的猜想，这个图$G'$必须是不连通的。本文给出了这个猜想的一个反例。

Reconstruction and Edge Reconstruction of Triangle-free Graphs

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs $\{G-v: v\in V(G)\}$. Let $diam(G)$ and $κ(G)$ denote the diameter and the connectivity of a graph $G$, respectively, and let $\mathcal{G}_2:=\{G: \textrm{diam}(G)=2\}$ and $\mathcal{G}_3:=\{G:\textrm{diam}(G)=\textrm{diam}(\overline{G})=3\}$. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in $\mathcal{G}_2\cup \mathcal{G}_3$. Balakumar and Monikandan showed that the Reconstruction …

排除不相交的 Kuratowski 图

Excluding disjoint Kuratowski graphs

本文研究不含 k 个互不相交 Kuratowski 子图的图结构，证明删除有界多个顶点后，剩余图可嵌入一个排斥所有 k-Kuratowski 图的曲面。

图重构中Kocay引理的推广与效力

Generalization and Power of Kocay's Lemma in Graph Reconstruction

该研究推广了Kocay引理并应用于图重构，探讨了推广引理与原引理的效力，证明了树作为图子图的重数可重构，丰富了图重构的理论工具。

A Simple Explanation for the Reconstruction of Graphs

The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of graphs. Here, we do not prove or reject the reconstruction conjecture. But, we explain why a graph is reconstructible. For instance, the reconstruction of small graphs which have been shown by computer, is explained in this framework. We show that any non-regular graph has a proper induced subgraph which is unique due to either its structure or the way of its …

$G$ -图的重构

$G$-reconstruction of graphs

设$G$是作用在$V$上的一个$n$ -顶点集上的一组置换，$X$和$Y$是$V$上的两个单图.若$Y$在$G$作用下属于$X$的轨道，则称$X$与$Y$是$G$ -同构的.人们可以自然地推广重建问题，使得当$G$是$S_n$，即对称群时，我们有通常的重建问题。本文研究了图的$G$ -边可重构性。证明了边重构和端点消去子图重构的一些新老结果。

Reconstruction of small graphs and digraphs

We describe computer searches that prove the graph reconstruction conjecture for graphs with up to 13 vertices and some limited classes on larger sizes. We also investigate the reconstructibility of tournaments up to 13 vertices, digraphs up to 9 vertices, and posets up to 13 points. In all cases, our results also apply to the set reconstruction problem that uses the isomorph-reduced deck.

Paley图及其推广图上的凝聚Ricci曲率

Condensed Ricci Curvature on Paley Graphs and their Generalizations

本研究探究广义Paley图性质，推广了Lim和Praeger的相关结果，给出素子域乘法群包含于非零k次幂乘法子群的充要条件，开发相关排序算法，推导出特定Paley图及其推广图上凝聚Ricci曲率的显式公式。

有界半径和直径的二分图上的匹配割及其变体

Matching Cut and Variants on Bipartite Graphs of Bounded Radius and Diameter

该研究探究匹配割及其多个变体问题在有界半径和直径二分图上的计算复杂度，拓展了已有结论，给出部分问题的复杂度二分法，解决了一个未决情形并得出首个难度结果。

独立数为三的图的独立吸引子的连通性

Connectedness of independence attractors of graphs with independence number three

本文研究独立数为三的图的独立吸引子的连通性，根据独立多项式系数 $a_1,a_2,a_3$ 的条件，分类刻画了吸引子为连通、完全不连通或不连通但非完全不连通的情形，并给出示例。

最大匹配割的二分性：$H$-无关性、有界直径、有界半径

Dichotomies for Maximum Matching Cut: $H$-Freeness, Bounded Diameter, Bounded Radius

研究 Maximum Matching Cut 的计算复杂性，在有界直径、有界半径和无 $H$ 图上给出完整二分性，并据此证明 Disconnected Perfect Matching 在无 $(P_6+sP_2)$ 图上多项式可解。

Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms

The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and local-global graph limit theories for sparse graphs. The natural limit objects are continuous actions of finitely generated groups on totally disconnected compact metric spaces. We prove that the space of weak equivalent classes of free Cantor actions is compact and contains a smallest element, as in the measurable case. We will introduce and study various notions …

图与重构算法不变量

Invariants algébriques de graphes et reconstruction

我们报告结果的代数图不变量的研究，基于计算机的探索，图形同构和重建问题的动机。

Rigidity and reconstruction for graphs

We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths".

随机图上模型的重构

Reconstruction for models on random graphs

考虑一个附着在图的顶点上的随机变量的集合。重建问题需要估计其中一个给定的“遥远”的观察。当它们的联合概率分布是关于树的马尔可夫时，几个理论结果（和简单算法）是可用的。在本文中，我们考虑的情况下，序列的随机图，局部收敛到树。特别是，我们开发了一个充分条件的树和图的重建问题相吻合。我们把这个条件应用到随机图的着色上。此外，我们的伊辛模型在这样的图形的行为特征，无论是有吸引力的和随机的相互作用（分别为“铁磁”和“自旋玻璃”）。