Related papers: On $k$-connectivity and minimum vertex degree in r…

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…

The minimum co-degree threshold for a perfect matching in a $k$-graph with $n$ vertices was determined by R\"odl, Ruci\'nski and Szemer\'edi for the case when $n\equiv 0\pmod k$. Recently, Han resolved the remaining cases when $n \not\equiv…

The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \cdots, T_k$ connecting…

Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…

For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a…

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 $G$…

Consider a random geometric graph defined on $n$ vertices uniformly distributed in the $d$-dimensional unit torus. Two vertices are connected if their distance is less than a "visibility radius" $r_n$. We consider {\sl Bluetooth networks}…

We consider the following definition of connectivity in $k$-uniform hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We determine the…

Halin showed that every edge minimal, k-vertex connected graph has a vertex of degree k. In this note, we prove the analogue to Halin's theorem for edge-minimal, k-edge-connected graphs. We show there are two vertices of degree k in every…

A connected graph is 2K2-free if it does not contain a pair of independent edges as an induced subgraph. In this paper, we present the structural characterization of minimal vertex separator and show that there are polynomial number of…

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal…

We study the properties of the giant connected component in random graphs with arbitrary degree distribution. We concentrate on the degree-degree correlations. We show that the adjoining nodes in the giant connected component are correlated…

An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$…

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

Extremal problems concerning the number of complete subgraphs have a long story in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in a graph $G$ and $m={{r_m}\choose s}+t_m$, where $0\le t_m\leq r_m$. Edr\H{o}s showed that…

We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…

A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…