We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a ≥ 2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u) + d(v) ≥ 3a whenever uv / ∈ A(D) and vu / ∈ A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D ...

Graph labeling traces its origin to the famous conjecture that all trees are graceful presented by A.Rosa in 1966. In recent years, many research papers on graceful graph have been published. In this survey we have collected studies on the gracefulness of the directed circuit ~ Cm and the union graph of mutually non-intersecting ~ Cm, the digraph n · ~ Cm, the digraph n − ~ Cm, the union of the...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l ≤ 2c− 1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We s...

The Gallai-Milgram theorem says that the vertex set of any digraph with stability number k can be partitioned into k directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer k, there is a digraph D with stability number k such that deleting the vertices of any k− 1 directed paths in D leaves a digraph with ...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...

A digraph is a symmetric cycle if it is symmetric and its underlying graph is a cycle. It is proved that if D is an asymmetric digraph not containing a symmetric cycle, then D remains asymmetric after removing some vertex. It is also showed that each digraph D without a symmetric cycle, whose underlying graph is connected, contains a vertex which is a common fixed point of all automorphisms of D.

We characterise the primitive 2-closed groups G of rank at most four that are not automorphism group a graph or digraph and show if degree is least 2402 then there just two infinite families G⩽AΓL1(pd), 1-dimensional affine semilinear group. These first known examples non-regular digraph.

An acyclic decomposition of a digraph is a partition of the edges into acyclic subgraphs. Trivially every digraph has an acyclic decomposition into two subgraphs. It is proved that for every integer sX2 every digraph has an acyclic decomposition into s subgraphs such that in each subgraph the outdegree of each vertex v is at most degðvÞ s 1 l m : For all digraphs this degree bound is optimal. r...