Berge’s elegant strong path partition conjecture from 1982 extends the Greene-Kleitman Theorem and Dilworth’s Theorem for all digraphs. The conjecture is known to be true for all digraphs for k = 1 by the Gallai-Milgram Theorem, and for k > 1 only for acyclic digraphs. We present a simple algorithmic proof for k = 1 which naturally extends to a new algorithmic proof for acyclic digraphs for all...

In this note we present a general approach to construct large digraphs from small ones. These are called expanded digraphs, and, as particular cases, we show the close relationship between lifted digraphs of voltage digraphs and line digraphs, which are two known ways to obtain dense digraphs. In the same context, we show the equivalence between the vertex-splitting and partial line digraph tec...

A primitive digraph D is said to be well primitive if the local exponents of D are all equal. In this paper we consider well primitive digraphs of two special types: digraphs that contain loops, and symmetric digraphs with shortest odd cycle of length r. We show that the upper bound of the exponent of the well primitive digraph is n− 1 in both these classes of digraphs, and we characterize the ...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V (D)− N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of...

It has been proved that sphericity testing for digraphs is an NP-complete problem. Here, we investigate sphericity of 3-connected single source digraphs. We provide a new combinatorial characterization of sphericity and give a linear time algorithm for sphericity testing. Our algorithm tests whether a 3-connected single source digraph with $n$ vertices is spherical in $O(n)$ time.

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

We consider 2-colored digraphs of the primitive ministrong digraphs having given exponents. In this paper we give bounds for 2-exponents of primitive extremal ministrong digraphs.

We construct a family of Cayley digraphs of degree d, diameter k and order kbd/2ck for any d ≥ 4 and k ≥ 3. We also present a collection of bipartite Cayley digraphs of order at least (k − 1)bd/2ck−1 for any degree d ≥ 4 and diameter k ≥ 4. For sufficiently large d and k, our digraphs are the largest known Cayley digraphs of degree d and diameter k, where k 6= d − 1 or d, and our bipartite digr...

Aharoni, R. and I. Ben-Arroyo Hartman, On Greene-Kleitman’s theorem for general digraphs, Discrete Mathematics 120 (1993) 13-24. Linial conjectured that Greene-Kleitman’s theorem can be extended to general digraphs. We prove a stronger conjecture of Berge for digraphs having k-optimal path partitions consisting of ‘long’ paths. The same method yields known results for acyclic digraphs, and exte...

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphsMCK(d, ) and it is derived from the Kautz...