Related papers: Extremality and Sharp Bounds for the $k$-edge-conn…

Driven by many applications in graph analytics, the problem of computing $k$-edge connected components ($k$-ECCs) of a graph $G$ for a user-given $k$ has been extensively studied recently. In this paper, we investigate the problem of…

Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S.…

We study the $k$-edge connectivity problem on undirected graphs in the distributed sketching model, where we have $n$ nodes and a referee. Each node sends a single message to the referee based on its 1-hop neighborhood in the graph, and the…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…

The $k$-independence number of a graph $G$ is the maximum size of a set of vertices at pairwise distance greater than $k$. In this paper, for each positive integer $k$, we prove sharp upper bounds for the $k$-independence number in an…

We consider the (exact, minimum) $k$-cut problem: given a graph and an integer $k$, delete a minimum-weight set of edges so that the remaining graph has at least $k$ connected components. This problem is a natural generalization of the…

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on…

The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…

A simple graph $G$ with edge-connectivity $\lambda(G)$ and minimum degree $\delta(G)$ is maximally edge connected if $\lambda(G)=\delta(G)$. In 1964, given a non-increasing degree sequence $\pi=(d_{1},\ldots,d_{n})$, Jack Edmonds showed…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…

Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if \( G - S \) is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for…

In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s_1,t_1}, ..., {s_k,t_k} in G, not necessarily distinct, there are…

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$…