Related papers: On the structure of oriented graphs and digraphs w…
Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B,…
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…
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse…
We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an…
Suppose one needs to change the direction of at least $\epsilon n^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least…
Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…
We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives…
We develop a basic theory for divisible design graphs with possible selfloops (LDDG's), and describe two infinite families of such graphs, some members of which are also classical examples of divisible design graphs without loops (DDG's).…
In this note, extending some results of Erdos, Frankl, Rodl, Alexeev, Bollobas and Thomason we determine asymptotically the number of graphs which do not contain certain large subgraphs. In particular, if H_1,...,H_n,... are graphs with…
In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and…
In a series of three papers we develop an end space theory for directed graphs. As for undirected graphs, the ends of a digraph are points at infinity to which its rays converge. Unlike for undirected graphs, some ends are joined by limit…
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,…
Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you…
Nash-Williams' Strong Immersion Conjecture states that graphs are well-quasi-ordered by the strong immersion relation. That is, given infinitely many graphs, one graph contains another graph as a strong immersion. In this paper we study the…
This work studies the typical structure of sparse $H$-free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph $H$. Extending the seminal result of Osthus, Pr\"omel, and Taraz that addressed the case where $H$…
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very…
A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds…
Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…
We prove that for every $\varepsilon > 0$ there exists $n_0=n_0(\varepsilon)$ such that every regular oriented graph on $n > n_0$ vertices and degree at least $(1/4 + \varepsilon)n$ has a Hamilton cycle. This establishes an approximate…
In this paper we initiate a systematic study of the Tur\'an problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the…