Strengthened chain theorems for different versions of 4-connectivity

Flow Extensions and Group Connectivity with Applications

We study the flow extension of graphs, i.e., pre-assigning a partial flow on the edges incident to a given vertex and aiming to extend to the entire graph. This is closely related to Tutte's $3$-flow conjecture(1972) that every $4$-edge-connected graph admits a nowhere-zero $3$-flow and a $\mathbb{Z}_3$-group connectivity conjecture(3GCC) of Jaeger, Linial, Payan, and Tarsi(1992) that every $5$-edge-connected graph $G$ is $\mathbb{Z}_3$-connected. Our main results show that these conjectures are equivalent to their natural flow extension versions and present some applications. The $3$-flow cas…

Connectivity and other invariants of generalized products of graphs

Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $Γ$ be a family of digraphs such that $V(F)=V$ for every $F\in Γ$. Consider any function $h:E(D)\longrightarrowΓ$. Then the product $D\otimes_{h} Γ$ is the digraph with vertex set $V(D)\times V$ and $((a,x),(b,y))\in E(D\otimes_hΓ)$ if and only if $(a,b)\in E(D)$ and $(x,y)\in E(h (a,b))$. In this paper, we introduce the undirected version of the $\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recover a generalization of the c…

On resilience of connectivity in the evolution of random graphs

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…

Zero-one laws for connectivity in random key graphs

The random key graph is a random graph naturally associated with the random key predistribution scheme of Eschenauer and Gligor for wireless sensor networks. For this class of random graphs we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to wireless sensor networks.

Tverberg plus constraints

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique that combines a concept of "Tverberg unavoidable subcomplexes" with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Zivaljevic and Vrecica (1992). We also get a new strengthened version of the…

Connectivity for quantum graphs

In quantum information theory there is a construction for quantum channels, appropriately called a quantum graph, that generalizes the confusability graph construction for classical channels in classical information theory. In this paper, we provide a definition of connectedness for quantum graphs that generalizes the classical definition. This is used to prove a quantum version of a particular case of the classical tree-packing theorem from graph theory. Generalizations for the related notions of $k$-connectedness and of orthogonal representation are also proposed for quantum graphs, and it i…

Extremality and Sharp Bounds for the $k$-edge-connectivity of Graphs

Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will…

铺位床猜想对仙人掌图及具有特定双连通分量的图仍然成立

The bunkbed conjecture still holds for cactus graphs and for graphs with certain biconnected components

本文区分铺位床猜想的强弱版本，证明图满足猜想当且仅当其所有双连通分量满足，并据此确认仙人掌图等图类成立，还给出强版本反例的 minor 必要条件。

通过连通支配集实现连通划分

Connected Partitions via Connected Dominating Sets

本文利用 k 个互不相交的连通支配集，在多项式时间内构造指定大小和顶点的连通划分，并据此得到一般图、凸二部图、双凸图和区间图上的 Győri–Lovász 构造性结果。

Connectivity Oracles for Graphs Subject to Vertex Failures

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_{\star} m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded …

平面强连通增强

Plane Strong Connectivity Augmentation

本文提出了一种固定参数可解算法，用于解决受平面性约束的强连通增强问题，该算法在时间上为2^{O(k)}n^{O(1)}。