Sebastian M. Cioabă, Xiaofeng Gu
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
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Asymptotic connectivity, hamiltonicity and toughness
本文提出连通性、哈密顿性、韧性和生成树的粗变体概念，这些概念在拟等距下不变，并保持经典关系，同时提出若干粗版本的开放问题。
Connectivity, toughness, and hamiltonicity of simplicial polytopes
针对 $d\ge4$ 的单纯多面体，构造了满足指定连通度、低坚韧度且不可迹的例子，否定了高连通度保证高维多面体哈密顿性的猜想，还得到了短度系数和指数的相关结论。
The $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{K_2,C_{2i+1}| i\geq 2 \}$-factor for any graph is obtained, which answers a problem due to Gao and Wang (J. Oper. Res. Soc. China (2021), https://doi.org/10.1007/s40305-021-00357-6).
For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and $$ c(G-S) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$} $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These results are …
l-connectivity, l-edge-connectivity and spectral radius of graphs
本文研究了简单图的l-连通性、l-边连通性与谱半径之间的联系，并探讨了有向图及边版本的类似问题，是对正则图基于第二大绝对特征值下界研究的拓展。
The \emph{prism} over a graph $G$ is the product $G \Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \geq 1$ a $P_4$-free graph has a spanning \emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\frac{1}{k}$-tough. As our main result, we sho…
The \textit{toughness} $t(G)$ of a graph $G$ is a measure of its connectivity that is closely related to Hamiltonicity. Brouwer proved the lower bound $t(G) > \ell / λ- 2$ on the toughness of any connected $\ell$-regular graph, where $ λ$ is the largest nontrivial eigenvalue of the adjacency matrix. He conjectured that this lower bound can be improved to $\ell / λ-1$ and this conjecture is still open. Brouwer also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly $\ell / λ$. Cioabă and W…
Spectral radius, toughness and $k$-factor of graphs
本文研究了连通1-韧性图中k-因子的存在条件，基于谱半径提出并证明了当δ(G)≥k时，存在k≥3的k-因子的充分条件。
Toughness and existence of $2$-factors
本文针对韧性在1到2之间的图，给出了存在2-因子的新的紧充分条件，条件涉及独立数、最小度、连通性和禁止森林，补充了Enomoto等人的结果。
Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles
1974年，Goodman和Hedetniemi证明了任何2-连通无$(K_{1,3},K_{1,3}+e)$图都是Hamilton图。这一结果在附加连通性条件下给出了各种禁止连通子图对和禁止连通子图三重图的其他哈密尔顿性条件。1997年证明了2-连通图中的单个禁止连通子图$R$只能生成一类平凡的具有$R=P_3$的哈密顿图（完全图）。本文证明了：如果$R$是不连通的，则一个禁止子图$R$可以生成一个非平凡的哈密顿图类：$(\ast1)$：每个无$(K_1\cup P_2)$的图都是Hamilton图或属于一个定义良好的非哈密顿图类; $(\ast2)$：每个无1-tough $(K_1\cup P_3)$的图都是Hamilton图.我们证明了每个1-tough $(K_1\cup P_4)$ -free图是Hamilton图，每个1-tough $P_4$ -free图是Hamilton图
We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.
Toughness Bounds for Fractional Hamiltonicity and Resistance Positivity
本文通过构造非哈密顿弦生成超图，结合弦图的哈密顿性定理，证明10-韧性图为分数哈密顿图，进而证实Devriendt关于10-韧性图为电阻正图的猜想，同时构造出韧性接近3/2的非电阻非负图。
We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.
Maximum number of spanning trees and connectivity: Graphs with a fixed minimum degree and bipartite graphs
本文研究了具有固定最小度数的图和双图在最大生成树数目方面的特性，确定了具有特定连通性和阶数的图结构。
The Complexity of Mixed Arc-Disjoint Spanning Subdigraphs with Antistrong Connectivity
该研究证明了关于反强连通性的两个混合弧不交生成子图判定问题均为NP-完全，还给出了其在特定有向图类上的困难性结果，否定了多项式时间求解的可能性。
A simple graph $G=(V,E)$ on $n$ vertices is said to be recursively partitionable (RP) if $G \simeq K_1$, or if $G$ is connected and satisfies the following recursive property: for every integer partition $a_1, a_2, \dots, a_k$ of $n$, there is a partition $\{A_1, A_2, \dots, A_k\}$ of $V$ such that each $|A_i|=a_i$, and each induced subgraph $G[A_i]$ is RP ($1\leq i \leq k$). We show that if $S$ is a vertex cut of an RP graph $G$ with $|S|\geq 2$, then $G-S$ has at most $3|S|-1$ components. Moreover, this bound is sharp for $|S|=3$. We present two methods for constructing new RP graphs from ol…
On the relation between connectivity, independence and generalized caterpillars
生成广义毛毛虫是指所有度大于2的顶点都在一条路上的生成树.本文讨论了生成广义毛毛虫的存在性与图的独立性和连通数之间的关系.本文还指出了L。Gargano等人在离散数学（2004）中的一个定理，并找出了另一个定理与生成广义毛虫的存在性之间的关系。
A $k$-container $C(u, v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^*$-connected if there exists a $k^*$-container between any two different vertices of G. A $k$-regular graph $G$ is super spanning connected if $G$ is $i^*$-container for all $1\le i\le k$. In this paper, we prove that the arrangement graph $A_{n, k}$ is super spanning connected if $n\ge 4$ and $n-k\ge 2$.
Spectral radius and edge-disjoint spanning trees of graphs with edge connectivity
本文解决了Zhang与Fan提出的关于k+1≤s≤2k-1的s-边连通图的紧谱半径条件及极图刻画问题，明确了图的边不交生成树最大数目与谱半径的关联。
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly gi…
An $O(k\log(n/k))$ Bound on Spanning Bipartite Connectivity
本文证明了任意高连通图包含生成双部 $k$-连通子图所需的最小连通度 $f(k,n)$ 满足 $f(k,n)=O(k\log(n/k))$，改进了先前结果，并部分验证了 Thomassen 猜想。
Scattering number and $τ$-toughness in graphs involving $A_α$-spectral radius
本文利用$A_α$-谱技术与结构分析，给出图满足$s(G)\leq1$及具有$τ$-韧性的谱半径充分条件，并推广和统一了已有结果。
Binding Number, Toughness and General Matching Extendability in Graphs
本文研究图中绑定数与韧度条件对一般匹配可扩展性的影响，证明在足够大的图中，若绑定数不低于4/3+ε或韧度不低于1+ε且高连通，则图具有E(m,n)-可扩展和(n,k)-可扩展性。
A New Perspective on Vertex Connectivity
边连通度和点连通度是图论中的两个基本概念。虽然到目前为止，基于图的边连通性对图的结构有了很好的理解，但我们在顶点连通性方面的知识要有限得多。在捕捉边连接的一个重要工具是边不相交的生成树。Tutte和Nash-Williams的著名结果表明，边连通的图$λ$包含$\floor{λ/2}$边不交生成树.我们提出了连接支配集（CDS）的划分和包装的工具，类似于边不相交的生成树，并帮助我们更好地掌握基于其顶点连通图的结构。CDS划分问题的目标是将图的节点划分为尽可能多的连通支配集。CDS包装问题是相应的分数松弛，其中CDS被允许重叠，只要通过分配适当的权重来补偿。CDS划分和CDS包装可以被看作是边不相交生成树的对应物，重点是顶点不相交而不是边不相交。我们构造性地证明了每一个具有$n$节点的$k$ -顶点连通图都有一个大小为$Ω(k/\log n)$的CDS填充和一个大小为$Ω(k/\log^5 n)$的CDS划分.我们证明了$Ω(k/\log n)$ CDS packing界是存在最优的。利用CDS填充，证明了如果$k$ -点连通图的顶点以概率$p$独立采样，则由采样点导出的图具有点连通性$\tildeΩ(kp^2)$ .此外，使用我们的$Ω(k/\log n)$ CDS包装，我们得到了一个存储和转发广播算法的最佳吞吐量的网络模型中，在每一轮，每个节点可以发送一个有限大小的消息到所有的邻…
Generalizing well-known results of Erdős and Lovász, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$, where $λ(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that…
Packing of Rigid Spanning Subgraphs and Spanning Trees
证明了每个（6 k + 2l，2k）-连通简单图都含有k个刚性l个连通边不交生成子图.这蕴涵了Jackson和Jordán [4]的一个定理和Jordán [6]关于刚性生成子图的填充的一个定理。这两个结果都是Lovász和Yemini [9]的经典结果的推广，即每个6-连通图都是刚性的，我们的方法提供了一个透明的证明。我们的结果还给出了两个改进的图的连通度上界，它们具有有趣的性质：（1）每个8-连通图都有一个生成树和一个2-连通生成子图;（2）每个14-连通图都有一个2-连通定向图.
We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)} f(e) =0$. We study the following type of questions, in several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee the existence of a zero-sum spanning tree of $G$? The types of $G$ we consider are complete graphs, $K_3$-free graphs, $d$-trees, and maximal planar graphs. We also answer the question of when any such colouring…
In this paper we study some open questions related to the smallest order $f({\cal C},\lnot {\cal H})$ of a 4-regular graph which has a connectivity property ${\cal C}$ but does not have a hamiltonian property ${\cal H}$. In particular, ${\cal C}$ is either connectivity, 2-connectivity or 1-toughness and ${\cal H}$ is hamiltonicity, homogeneously traceability or traceability. A standard theoretical approach to these questions had already been used in the literature, but did not succeed in determining the exact value of $f()$. Here we have chosen to use Integer Linear Programming and to encode t…
Characterizations of minimal graphs with equal edge connectivity and spanning tree packing number
由于图被认为是许多网络设计问题的自然模型，图$G$的边连通度$κ'(G)$和边不相交生成树的最大数量$τ(G)$已经被用作对建模为图$G$的通信网络中的可靠性和强度的度量（参见\cite{Cunn85, Matula87}等）。Mader \cite{Mader71}和Matula \cite{Matula72}引入了最大子图边连通度$\overline{κ'}(G)=\max \{κ'(H): H \mbox{ is a subgraph of } G \}$。基于它们在网络设计中的应用和已建立的不等式\[ \overline{κ'}(G)\ge κ'(G) \ge τ(G), \]，本文给出了以下结果：（i）对于每个整数$k>0$，给出了具有$\overline{κ'}(G) \le k$性质的图$G$的一个特征，但对于任何不在$G$中的边$e$，$\overline{κ'}(G+e)\ge k+1$ . (ii)对任意整数$n > 0$，给出了具有$|V(G)| = n$的图$G$使得具有$|E(G)|$的$κ'(G) = τ(G)$最小化的一个刻划.
Let $τ(G)$ and $κ'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $δ\ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $λ_2(G) < δ- \frac{2k-1}{δ+1}$, then $τ(G) \ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(δ, k, g)$ such that for every graph $G$ with minimum degree $δ\ge 2k \ge…
Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivit…
Spectral and size conditions for spanning k-trees in tough graphs
针对韧度在[1/k, 1/(k-1))范围内的图，利用谱半径和无符号拉普拉斯谱半径给出存在生成k-树的充分条件，并建立边数下界。
The polynomial algorithm for optimal spanning hyperforest problem
本文研究了超图连通性的一个理论，并证明了对于任意给定的加权q-一致超图，求最优生成超森林（超树）的多项式算法。
An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts of size less than $k$. If $r=1$, we simply call it essential. Recently, Lai and Li proved that every $m$-edge-connected essentially $h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where $k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge \frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and $2$-essen…
Robust Connectivity of Graphs on Surfaces
本文研究了嵌入曲面中的图的鲁棒连通性，证明了r-连通图的鲁棒连通性在欧拉亏量为γ时的渐进行界，并揭示了图的鲁棒连通性与曲面连通性之间的联系。
Connectivity preserving spanning $(u,v)$-paths in $k$-connected graphs
本文证明：对于最小度至少为 (n+1)/2 的 k-连通图 G，存在哈密顿 (u,v)-路 P 使得 G-E(P) 仍 k-连通，并推广到 s 条内部不相交的生成 (u,v)-路。
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Γ$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Γ$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$,…
The generalized connectivity of complete bipartite graphs
设$G$是一个阶为$n$的非平凡连通图，$k$是一个具有$2\leq k\leq n$的整数.对于$G$的$k$顶点的集合$S$，令$κ(S)$表示$G$中的边不相交树$T_1,T_2,...,T_\ell$的最大数目$\ell$，使得对于具有$1\leq i,j\leq \ell$的每个不同整数对$i,j$，$V(T_i)\cap V(T_j)=S$。Chartrand等人将连通性的概念概括如下：$G$的$k$-$connectivity$，用$κ_k(G)$表示，由$κ_k(G)=$ min $\{κ(S)\}$定义，其中最小值取$V(G)$的所有$k$ -子集$S$。因此，$κ_2(G)=κ(G)$，其中$κ(G)$是$G$的连通性。此外，$κ_{n}(G)$是$G$的边不相交生成树的最大数目。本文主要研究了完全二部图PDMATHBNF的PDMATHBMO-连通性.首先，我们得到了$K_{a,b}$的边不交生成树的个数，即$\lfloor\frac{ab}{a+b-1}\rfloor$，并具体给出了$\lfloor\frac{ab}{a+b-1}\rfloor$边不交生成树。在此基础上，我们得到了所有$2\leq k \leq a+b$的$K_{a,b}$的$k$连通性.即，如果$k>b-a+2$和$a-b+k$是奇数，则$κ_{k}(K_{a,b})=\frac{a+b-k…
On the connectivity Waiter-Client game
在这个简短的说明中，我们考虑一个变化的连接Waiter-Client游戏$WC(n,q,\mathcal{A})$上播放的$n$ -顶点图$G$，其中包括$q+1$不相交的生成树。在这个游戏中，在每一轮服务员提供客户端$q+1$边缘的$G$尚未提供。客户端选择一条边，其余的$q$边将被丢弃。Waiter的目的是迫使Client构建一个连通图。如果发生这种情况，服务员赢了。否则，客户就是赢家。我们考虑的情况下，$2 < q+1 < \lfloor \frac{n-1}{2}\rfloor$，并表明，对于每个这样的$q$存在一个图$G$，客户端有一个获胜的策略。这个结果与$G$仅由2个生成树组成或$G$是完全图的情况相反，因为已经表明，对于这样的图，Waiter总是可以迫使Client构建连通图。
Sharp connectivity thresholds for mixed rigidity packings and improved bounds for highly connected orientations of graphs
本文证明了一个统一定理，给出混合刚性生成子图填充的尖锐连通性阈值，并改进了高度连通定向的二次上界，解决了两个猜想并降低了首项系数。
Packing spanning trees and the $k$-tree protocol
本文给出了最大生成树可包装图的结构描述和不变量，即G的边连通度等于G中边不相交生成树的最大个数的图G.这些图对于Itai和Rodeh [Inform.的Comput。79（1988），43-59]。
A note on packing spanning trees in graphs and bases in matroids
我们考虑一类图的边连通度等于边不相交的生成树的最大数目，并自然推广到拟阵，其中cogirth等于不相交的基的数目。我们提供这样的图和拟阵的描述，表明这样的图（或拟阵）有一个独特的分解。在图的情况下，我们的结果是相关的某些通信协议。
k-Edge-Connectivity: Approximation and LP Relaxation
在k-边连通生成子图问题中，我们给定一个图（V，E）和每条边的代价，并希望找到E的最小代价子集F，使得（V，F）是k-边连通的。我们证明了存在一个常数eps > 0，因此对于所有k > 1，找到k-ECSS的（1 + eps）近似是NP困难的，在单位成本和一般成本版本之间建立了一个差距。接下来，我们考虑k-ECSS的多子图表亲，其中我们购买E的多子集F，以与原始边相同的成本提供无限的并行副本。我们猜想存在一个（1 + Theta（1/k））-近似算法，并描述了一种基于图分解的自然线性规划（LP）松弛方法。该LP基本上等价于TSP的Held-Karp LP和Steiner树的无向LP。我们给出了一个家庭的极端点的LP是比以前已知的更复杂。
Color-avoiding connected spanning subgraphs with minimum number of edges
本文研究避色连通图的最大可移除边数问题，证明其为NP难问题，提出多项式时间近似算法，确定了给定参数下避色连通图的最小边数，并将边避色连通性推广至拟阵。
Sufficient conditions for spanning $k$-trees in tough graphs
研究在坚韧图中生成k-树的充分条件，通过谱条件和图的坚韧度分析，提出新的谱半径和无向拉普拉斯谱半径的充分条件。
Approximation and FPT Algorithms for Finding DM-Irreducible Spanning Subgraphs
本文针对与二分图Dulmage--Mendelsohn分解相关的DM-不可约生成子图推广问题，拓展了原有向图强连通生成子图问题的2-近似算法及固定参数可处理结果。
An Improved Degree Condition for Connectivity-Preserving Spanning $(u,v)$-Paths
本研究针对Teng和田提出的关于k-连通图中保持连通性的生成(u,v)-路径的最小度条件猜想，给出了肯定证明，同时改进了图的阶数下界要求。
Connectivity Graph-Codes
本文定义了图 $H$ 的连通性码，证明了 $d$-正则扩展图的最大连通性码基数达到上界 $2^{δ(H)}$，并指出在三次图等自然实例中该等号不成立。
An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube
该研究解决了四维对偶立方F₄是否存在两棵完全独立生成树的问题，通过显式构造完成了Fₙ存在该树的n≥4的分类，还得出了相关规则的不可行性结果。
Completely Independent Steiner Trees
本文研究完全独立的Steiner树，提出了一种新的树结构，要求任意两个终端之间的路径在顶点和边方面均不相交，扩展了完全独立的生成树和内部不相交的Steiner树概念，并探讨了其结构、算法及复杂性理论。
A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes
本文通过组合学和电信网络方法证明了关联方案中图的边连通度等于其顶点度数，并展示了连通正则等arboreal图具有完美匹配的性质。
A $5/4$-Approximation for Two-Edge Connectivity
针对二边连通生成子图问题，提出5/4近似算法，运行时间为多项式，优于先前1.3+ε近似比。
Connectivity of the reconfiguration graph of Hamiltonian paths
本文研究哈密顿路径重构图$R(G)$的连通性，证明当$G$包含哈密顿路径的平方作为生成子图时$R(G)$连通且可用$O(n^2)$次回咬操作重构，并构造了高连通但重构图不连通的例子。
The spanning tree packing number of a graph $G$, denoted by $τ(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $τ(G)$ is one of the classic problems in graph theory. Cioabă and Wong initiated to investigate $τ(G)$ from spectral perspectives in 2012 and since then, $τ(G)$ has been well studied using the second largest eigenvalue of the adjacency matrix in the past decade. In this paper, we further extend the results in terms of the number of edges and the spectral radius, respectively; and prove tight sufficient conditions to guarantee $τ(G)\geq k$ with…
In this note we establish a resilience version of the classical hitting time result of Bollobás and Thomason regarding connectivity. A graph $G$ is said to be $α$-resilient with respect to a monotone increasing graph property $\mathcal{P}$ if for every spanning subgraph $H \subseteq G$ satisfying $\mathrm{deg}_H(v) \leq α\cdot \mathrm{deg}_G(v)$ for all $v \in V(G)$, the graph $G - H$ still possesses $\mathcal{P}$. Let $\{G_i\}$ be the random graph process, that is a process where, starting with an empty graph on $n$ vertices $G_0$, in each step $i \geq 1$ an edge $e$ is chosen uniformly at ra…
The basic goal of survivable network design is to construct low-cost networks which preserve a sufficient level of connectivity despite the failure or removal of a few nodes or edges. One of the most basic problems in this area is the $2$-Edge-Connected Spanning Subgraph problem (2-ECSS): given an undirected graph $G$, find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges (in particular, $H$ remains connected after the removal of one arbitrary edge). 2-ECSS is NP-hard and the best-known (polynomial-time) approximation factor for this problem is $4/3$. Interest…
Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rflo…
Spanning Structures in Walker--Breaker Games
本文研究偏差为 $(2:b)$ 的 Walker--Breaker 博弈，证明在连通性博弈和哈密顿圈博弈中，即使 Breaker 偏差达 $n/\ln n$，Walker 仍能获胜。
On sufficient conditions for spanning structures in dense graphs
该研究将稠密图中哈密顿图的鲁棒结构条件推广到环幂与次线性带宽图，解决了相关嵌入问题并恢复建立了多场景下的带宽定理。
On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs
本文研究随机图和随机扰动稠密图中生成树的颜色偏差，证明在连通阈值以上的随机图及添加少量随机边的稠密图中，任意2-边着色下都存在具有较大颜色偏差的生成树。
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