Related papers: Menger's theorem for ends of digraphs

We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex

Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $\delta$ and $p$ - the order of a…

It is proved that the number of shortest paths between two vertices of distance $t$ in a graph with degrees bounded by $\Delta$ is at most $2 \cdot (\frac{\Delta}{2})^t$. This improves upon the na\"ive $\Delta (\Delta-1) ^{t-1}$ bound.

An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in…

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the…

We introduce a new balanced separator theorem for unit-disk graphs involving two shortest paths combined with the 1-hop neighbours of those paths and two other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA '12] and…

Let $B$ be a bidirected multigraph with signing $\sigma$, let $X$ be a set of vertices in $B$, and let $k$ be a non-negative integer. For any pair of vertex sets $S,T\subset V(B)$ satisfying $X\cap S = X\cap T$, we denote by $B_{S,T}$ the…

A separating path system for a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that for every two edges $e$ and $f$ in $G$, there is a path in $\mathcal{P}$ that contains $e$ but not $f$. We show that every $n$-vertex graph has…

A graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in each pair. We obtain sharp bounds on the maximum possible diameter of path-pairable graphs which either have a given number…

For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The $\theta_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $\theta_r$. This…

A celebrated theorem of Stiebitz asserts that any graph with minimum degree at least $s+t+1$ can be partitioned into two parts which induce two subgraphs with minimum degree at least $s$ and $t$, respectively. This resolved a conjecture of…

Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a group of…

The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$.…

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any depth-$k$…

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex…

We study a generalization of the well-known disjoint paths problem which we call the metric Menger problem, denoted MM(r,k), where one is given two subsets of a graph and must decide whether they can be connected by $k$ paths of pairwise…

A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated…

We prove the following theorem. Let $r\ge 4$ be an integer, and $G$ be a $K_{1,r}$-free $r$-edge-connected $r$-regular graph. Then, for every set $W$ of even number of vertices of $G$ such that the distance between any two vertices of $W$…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…