Related papers: Covering the edges of digraphs in $\mathscr{D}(3,3…
Given an exact covering system $S = \{a_i$ (mod $d_i$) $: 1 \leq i \leq r\}$, we introduce the corresponding exact covering system digraph (ECSD) $G_S = G(d_1n+a_1, \ldots, d_rn+a_r)$. The vertices of $G_S$ are the integers and the edges…
Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$.…
A particular case of Caccetta-H\"{a}ggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order…
We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum…
Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}.$ We improve the lower bound from $\sqrt{d}$ to…
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…
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…
A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results…
Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of…
For the Kautz digraph $K(d,D)$, let $\rho_k(d,D)$ be the number of oriented edges whose shortest directed cycle has length $k+1$, and define $\Delta_k(d,D) = \rho_k(d,D) - \rho_k(d,D-1)$. We give an exact, finite-dimensional matrix product…
For $\Delta$ a finite connected nontrivial directed multigraph, we prove: 1. $\Delta$ has a directed circuit using each directed edge exactly once if and only if both each pair of distinct vertices of $\Delta$ occur in a common directed…
Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$…
In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…
A directed hypergraph (dihypergraph) consists of a set of vertices and a set of hyperarcs, where each hyperarc is partitioned into a head and a tail. Directed hypergraphs are useful in many applications, including the study of chemical…
In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter $D$ and edge metric dimension $k$…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for…
An out-tree $T$ of a directed graph $D$ is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By $l(D)$ and $l_s(D)$ we denote the maximum number of leaves over all out-trees and…
An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In…
We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that…