We count labeled acyclic digraphs according to the number sources, sinks, and edges. 1. Counting acyclic digraphs by sources. Let An(t;α) = ∑ D αt, where the sum is over all acyclic digraphs D on the vertex set [n] = {1, 2, . . . , n}, e(D) is the number of edges of D, and s(D) is the number of sources of D; that is, the number of vertices of D of indegree 0. Let An(t) = An(t; 1). To find a rec...

We introduce the circular chromatic number χc of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k-cycle has circular chromatic number k/(k − 1), for k ≥ 2, values of χc between 1 and 2 are possible. We show tha...

A feedback arc set of a digraph is a set of arcs whose reversal makes the resulting digraph acyclic. Given a tournament with a disjoint union of directed paths as a feedback arc set, we present necessary and sufficient conditions for this feedback arc set to have minimum size. We will present a construction for tournaments where the difference between the size of a minimum feedback arc set and ...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when clustering, indexing and bounding space for various genetic data for efficient analysis. We discuss the corresponding chromatic number and derive an upper bound as a function of the maximum number of descend...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding chromatic number and derive an upper bound as a function of the maximum number of descendants of a given vertex an...

We consider vertex coloring of a simple acyclic digraph G in such a way that two vertices which have a common ancestor in G receive distinct colors. Such colorings arise in a natural way when clustering, indexing and bounding space for various genetic data for efficient analysis. We discuss the corresponding chromatic number and derive an upper bound as a function of the maximum number of desce...

In the thesis, the coloring of digraphs is studied. The chromatic number of a digraph D is the smallest integer k so that the vertices of D can be partitioned into at most k sets each of which induces an acyclic subdigraph. A set of four topics on the chromatic number is presented. First, the dependence of the chromatic number of digraphs on the maximum degree is explored. An analog of Gallai’s...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOMP(H). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism probl...

Given a digraph D and a subset X of vertices of D, pushing X in D means reversing the orientation of all arcs with exactly one end in X. It is known that the problem of deciding whether a given digraph can be made acyclic using the push operation is NP-complete for general digraphs, and polynomial time solvable for multipartite tournaments. Here, we continue the study of deciding whether a digr...