Related papers: Partitioning transitive tournaments into isomorphi…
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce…
A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various…
We initiate a general study of what we call orientation completion problems. For a fixed class C of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in…
Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This…
The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph.…
Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is…
A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks,…
A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be…
We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$ such that $G\not\subseteq K$, or (ii)…
It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection…
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to…
The modular decomposition of a graph is a canonical representation of its modules. Algorithms for computing the modular decomposition of directed and undirected graphs differ significantly, with the undirected case being simpler, and…
The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several…
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc…
An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge…
K\"uhn and Osthus conjectured in 2013 that regular tripartite tournaments are decomposable into Hamilton cycles. Somewhat surprisingly, Granet gave a simple counterexample to this conjecture almost 10 years later. In this paper, we show…
We say that a digraph $D$ is competitive if any pair of vertices has a common out-neighbor in $D$ and that a graph $G$ is competitively orientable if there exists a competitive orientation of $G$. The notion of competitive digraphs arose…
The converse of a tournament is obtained by reversing all arcs. If a tournament is isomorphic to its converse, it is called self--converse. Eplett provided a necessary and sufficient condition for a sequence of integers to be realisable as…
A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Br\"omssen (n\'e Andersson) showed that the existence of such an…
We prove that for every $n\geq 2$ the class of all finite $n$-partite tournaments (orientations of complete $n$-partite graphs) has the extension property for partial automorphisms, that is, for every finite $n$-partite tournament $G$ there…