Related papers: Constant congestion linkages in polynomially stron…

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum…

Loebl, Koml\'os, and S\'os conjectured that any graph with at least half of its vertices of degree at least k contains every tree with at most k edges. We propose a version of this conjecture for skewed trees, i.e., we consider the class of…

Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for all $G \in {\mathcal G}$ but finitely many, the following holds. Any…

Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want…

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there…

Suppose that $D=(V,E)$ is a strongly connected digraph. Let $u,v\in V(D)$. The maximum distance $md (u,v)$ is defined as $md(u,v)$=max\{$\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)$\} where $\overrightarrow{d}(u,v)$ denote the length…

In this paper, we investigate normal trees of directed graphs, which extend the fundamental concept of normal trees of undirected graphs. We prove that a directed graph $D$ has a normal spanning tree if and only if the topological space…

A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic…

We prove that the directed treewidth, DAG-width and Kelly-width of a digraph are bounded above by its circumference plus one.

Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2, \cdots, T_r$ in $G$ such that $V(T_i)\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with…

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected…

In this note, we study Laplacians on graphs for which connectivity within certain subgraphs tends to infinity. Our main focus are graphs sharing a common node set on which edge weights within certain clusters grow to infinity. As…

A set $X$ of vertices of an acyclic digraph $D$ is convex if $X\neq \emptyset$ and there is no directed path between vertices of $X$ which contains a vertex not in $X$. A set $X$ is connected if $X\neq \emptyset$ and the underlying…

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley…

Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph…

We show that the spanning tree congestion problem is NP-complete even for proper interval graphs of linear clique-width at most 4.

In 2001, Koml\'os, S\'ark\"ozy and Szemer\'edi proved that, for each $\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+\alpha)n$ contains a copy of every…

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if…

A directed graph $D$ is singly connected if for every ordered pair of vertices $(s,t)$, there is at most one path from $s$ to $t$ in $D$. Graph orientation problems ask, given an undirected graph $G$, to find an orientation of the edges…

In 2015, Kawarabayashi and Kreutzer proved the Directed Grid Theorem - the generalisation of the well-known Excluded Grid Theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour and Thomas from the…