The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.
The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce -- under the orientation mapping -- infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the $(2\ell+1)$-wheel graph with a nonzero coefficient at every $\ell\in\mathbb{N}$. We present detailed calculations of the differential of graphs; …
An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an oriented graph $G$ with $m$ edges, such that every $2$-colouring of $G$ contains a monochromatic copy of $H$. In this paper we prove that the oriented size Ramsey number of the directed paths on $n$ vertices satisfies $r(P_n) = Ω(n^2 \log n)$. This improves a lower bound by Ben-Eliezer, Krivelevich and Sudakov. It also matches an upper bound by Bucić and the au…
A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $γ(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $Γ_d(G)$, which is the maximum directed domination number $γ(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erdös [Math. Gaz. 47 (1963), 220--222], albeit in disguised fo…
Erdős [On Schütte problem, Math. Gaz. 47 (1963)] proved that every tournament on $n$ vertices has a directed dominating set of at most $\log (n+1)$ vertices, where $\log$ is the logarithm to base $2$. He also showed that there is a tournament on $n$ vertices with no directed domination set of cardinality less than $\log n - 2 \log \log n + 1$. This notion of directed domination number has been generalized to arbitrary graphs by Caro and Henning in [Directed domination in oriented graphs, Discrete Appl. Math. (2012) 160:7--8.]. However, the generalization to directed r-uniform hypergraphs seems…
A graph $G$ has a $k$-page book embedding if $G$ can be embedded into a $k$-page book. The minimum $k$ such that $G$ has a $k$-page book embedding is the book thickness of $G$, denoted $bt(G)$. Most of the work on this subject has been done for unoriented graphs and oriented acyclic graphs (no directed cycles). In this work we discuss oriented graphs $\overrightarrow{D}$ containing directed cycles by using oriented book embeddings and oriented book thickness, $obt(\overrightarrow{D})$. To characterize $\overrightarrow{D}$ such that $obt(\overrightarrow{D}) = k$, we define the class $\mathcal{M…
A particular case of Caccetta-Häggkvist 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 $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any …
Let $D$ be a strongly connected oriented graph with vertex-set $V$ and arc-set $A$. The distance from a vertex $u$ to another vertex $v$, $d(u,v)$ is the minimum length of oriented paths from $u$ to $v$. Suppose $B=\{b_1,b_2,b_3,...b_k\}$ is a nonempty ordered subset of $V$. The representation of a vertex $v$ with respect to $B$, $r(v|B)$, is defined as a vector $(d(v,b_1), d(v,b_2), ..., d(v,b_k))$. If any two distinct vertices $u,v$ satisfy $r(u|B)\neq r(v|B)$, then $B$ is said to be a resolving set of $D$. If the cardinality of $B$ is minimum then $B$ is said to be a basis of $D$ and the ca…
In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the…
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each non-directed cycle of order m, 2 < m < n+1.
Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed hypergraph on n vertices can have without containing a copy of F. There are actually two versions the directed hypergraph model for this problem: the standard version where every triple of vertices is allowed to have up to all three possible directed edges and the oriented version where each triple can have at most one directed edge. In this paper, we determine …
Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed hypergraph on n vertices can have without containing a copy of F. There are actually two versions of this problem: the standard version where every triple of vertices is allowed to have up to all three possible directed edges and the oriented version where each triple can have at most one directed edge. In this paper, we determine the standard extremal numbers a…
A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.
In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $ \overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Rams…
We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.
We study analogues of Sidorenko's conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures.
The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the under…
We introduce the fractional version of oriented coloring and initiate its study. We prove some basic results and study the parameter for directed cycles and sparse planar graphs. In particular, we show that for every $ε> 0$, there exists an integer $g_ε \geq 12$ such that any oriented planar graph having girth at least $g_ε$ has fractional oriented chromatic number at most $4+ε$. Whereas, it is known that there exists an oriented planar graph having girth at least $g_ε$ with oriented chromatic number equal to $5$. We also study the fractional oriented chromatic number of directed cycles and pr…
In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented bond matroi…
To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices. …
Given a graph $H$, the size Ramsey number $r_e(H,q)$ is the minimal number $m$ for which there is a graph $G$ with $m$ edges such that every $q$-coloring of $G$ contains a monochromatic copy of $H$. We study the size Ramsey number of the directed path of length $n$ in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors, showing that for every $q\geq 1 $ there are constants $c_1 = c_1(q),c_2$ such that $$\frac{c_1(q) n^{2q}(\log n)^{1/q}}{(\log\log n)^{(q+2)/q}} \leq r_e(\overrightarrow{P_n},q+1) \leq c_2 n^{2q}(\log {n})^2.$$ O…
In this paper we obtain a new sufficient condition for the existence of directed cycles of length 4 in oriented bipartite graphs. As a corollary, a conjecture of H. Li is confirmed. As an application, a sufficient condition for the existence of rainbow cycles of length 4 in bipartite edge-colored graphs is obtained.
In 2001, Komlós, Sárközy and Szemerédi proved that, for each $α>0$, 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 $(1/2+α)n$ contains a copy of every $n$-vertex tree with maximum degree at most $cn/\log n$. We prove the corresponding result for directed graphs. That is, for each $α>0$, there is some $c>0$ and $n_0$ such that, if $n\geq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+α)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/\log n$. …
The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…
We survey some recent results on long-standing conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly's conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are `almost' edge-disjoint. We also highlight the role that the notion of `robust expansion' plays in several of the proofs. New and old open problems are discussed.
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle $C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even o…
We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of ``topologically t-chromatic'' graphs. We show that this minimum for large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski graphs of appropriate parameters is the upper integer part o…
Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has led to many results about threshold graphs and similar structures. We look to generalize this class of graphs to oriented graphs (directed simple graphs.) We give generalizations to four of the most versatile definitions and show their equivalence in the oriented case. We finish with a proof enumerating the number of these oriented threshold graphs which relat…
Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.
It is well known that a tournament (complete oriented graph) on $n$ vertices has at most ${1/4}\binom{n}{3}$ directed triangles, and that the constant 1/4 is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some `higher order' versions of this statement. For example, if we give each 3-set from an $n$-set a cyclic ordering, then what is the greatest number of `directed 4-sets' we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each $d$-set from an $n$-set.
A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $γ(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $Γ_d(G)$, which is the maximum directed domination number $γ(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erdös [Math. Gaz. 47 (1963), 220--222], albeit in disguised fo…
In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all preferred extensions can be enumerated in $O^*(3^{n/3})$ time and there are directed graphs with $Ω(3^{n/3})$ preferred extensions. We give faster enumeration algorithms for directed graphs with at most $0.8004\cdot n$ vertices occurring in $2$-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time $O(1.2321^n…
Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $λ(\vec C)=0$, where $λ(\vec C)$ is the number of forward arcs minus the number of backward arcs in $\vec C$. This is not true. In this paper we show that $\vec C$ can be colored with three colors if and only if $λ(\vec C)=0(\bmod~3)$ or $\vec C$ does not contain three consecutive arcs going in the same direction.
Let $G$ be a simple graph with an orientation $σ$, which assigns to each edge a direction so that $G^σ$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^σ$. In this paper, we define a weighted skew adjacency matrix with Randć weight, the skew Randić matrix ${\bf R_S}(G^σ)$, of $G^σ$ as the real skew symmetric matrix $[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-\frac{1}{2}}$ and $(r_s)_{ji} = -(d_id_j)^{-\frac{1}{2}}$ if $v_i \rightarrow v_j$ is an arc of $G^σ$, otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$. We derive some properties of the skew Randić energy…
Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs: - Erdős-Gallai-type results: characterization of net-degree sequences, - Havel-Hakimi-type results: complete sets of degree-preserving operations, - Extremal degree sequences: characterization of uniquely realizable sequences, and - Enumerative aspects: counting formulas for net-d…
A $(0,1)$-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let $g$ be a labeling of the edge set of a graph that is induced by a labeling $f$ of the vertex set. If both $g$ and $f$ are friendly then $g$ is said to be a cordial labeling of the graph. We extend this concept to directed graphs and investigate the cordiality of directed graphs. We show that all directed paths and all directed cycles are cordial. We also discuss the cordiality of oriented trees and other digraphs.
The randomly oriented graph $G_{n,p}^σ$ is an Erdős-Rényi random graph $G_{n,p}$ with a random orientation $σ$, which assigns to each edge a direction so that $G_{n,p}^σ$ becomes a directed graph. Denote by $S_n$ the skew-adjacency matrix of $G_{n,p}^σ$. Under some mild assumptions, it is proved in this paper that, the spectral distribution of $S_n$ (under some normalization) converges to the standard semicircular law almost surely as $n\rightarrow\infty$. It is worth mentioning that our result does not require finite moments of the entries of the underlying random matrix.
The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph $D$ is $3$-dicritical if the removal of any arc of $D$ reduces the dichromatic number to 2. We construct infinitely many $3$-dicritical oriented graphs. Neumann-Lara found the four $7$-vertex $3$-dichromatic tournaments. We determine the $8$-vertex $3$-dichromatic tournaments, which …
We prove that the inclusion from oriented graph complex into graph complex with at least one source is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph complex. This result may have applications in theory of multi-vector fields $T_{\rm poly}^{\geq 1}$ of degree at least one, and to the hairy graph complex which computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play …
