For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^k$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^k$ if the directed distance ${}^{\to }{d}_D(u,v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^k$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^k$ is distance-colored if each arc (u, v) of $D^k$ is assigned the color i where $i{=}^{\to }{d}_D(u,v)$. The digraph $D^k$ is Hamiltonian-colored...

Let $(H,r)$ be a fixed rooted digraph. The $(H,r)$-coloring problem is the problem of deciding for which rooted digraphs $(G,s)$ there is a homomorphism $f:G\to H$ which maps the vertex $s$ to the vertex $r$. Let $(H,r)$ be a rooted oriented path. In this case we characterize the nonexistence of such a homomorphism by the existence of a rooted oriented cycle $(C,q)$, which is homomorphic to $(G,s)$ but not homomorphic to $(H,r)$. Such a property of the digraph $(H,r)$ is called rooted cycle duality or $*$-cycle duality. This extends the analogical result for...

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring...

A general model of a random digraph D(n,P) is considered. Based on a precise estimate of the asymptotic behaviour of the distribution function of the binomial law, a problem of the distribution of extreme in-degrees of D(n,P) is discussed.

In 1982 Laborde, Payan and Xuong [Independent sets and longest directed paths in digraphs, in: Graphs and other combinatorial topics (Prague, 1982) 173-177 (Teubner-Texte Math., 59 1983)] conjectured that every digraph has an independent detour transversal (IDT), i.e. an independent set which intersects every longest path. Havet [Stable set meeting every longest path, Discrete Math. 289 (2004) 169-173] showed that the conjecture holds for digraphs with independence number two. A digraph is p-deficient...

We present several results concerning the Laborde-Payan-Xuang conjecture stating that in every digraph there exists an independent set of vertices intersecting every longest path. The digraphs we consider are defined in terms of local semicompleteness and local transitivity. We also look at oriented graphs for which the length of a longest path does not exceed 4.

Given a simple directed graph D = (V,A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let D ∈ D(n, p) (with p = p(n)) be a random instance, obtained by randomly orienting each edge of a random graph drawn from Ϟ(n, 2p). We show that mat(D) is asymptotically almost surely (a.a.s.) one of only 2 possible values, namely either b*or b* + 1, where b* = ⌊2(logrn) + 0.5⌋ and r = p−1. It is also shown that if, asymptotically, 2(logrn) + 1 is not within a distance of w(n)/(ln...

In this paper, we construct infinite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an infinite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show that any tournament with a tight mold is also tight.

Let $G$ be a simple graph. A function $f$ from the set of orientations of $G$ to the set of non-negative integers is called a continuous function on orientations of $G$ if, for any two orientations $O_1$ and $O_2$ of $G$, $|f\left(O_1\right)-f\left(O_2\right)|\le 1$ whenever $O_1$ and $O_2$ differ in the orientation of exactly one edge of $G$. We show that any continuous function on orientations of a simple graph $G$ has the interpolation property as follows: If there are two orientations $O_1$ and $O_2$ of $G$ with $f\left(O_1\right)=p$ and $f\left(O_2\right)=q$, where $p<q$, then for any integer $k$ such that $p<k<q$, there are...