Galeana-Sanchez, H., On the existence of kernels and h-kernels in directed graphs, Discrete Mathematics 110 (1992) 2.51-255. A directed graph D with vertex set V is called cyclically h-partite (h > 2) provided one can partition V = V, + V, +. . + V,_, so that if (u, u) is an arc of D then u E V,, and u E v+, (notation mod h). In this communication we obtain a characterization of cyclically h-pa...

A digraph is k ${\bf{k}}$ -strong if it has n ≥ + 1 $n\ge k+1$ vertices and every induced subdigraph on at least − $n-k+1$ strongly connected. tournament a with no pair of nonadjacent vertices. We prove that can be made $k$ by adding more than 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs. This solves conjecture from 1994. semicomplete there one arc between any distinct x , y $x,y$ . Since ...

Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering dened by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We p...

Let D be a digraph, and let L(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(L(D))= k(D)=0 if every connected component of D is a directed cycle. (2) k(L(D))= k(D)+ n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that ...

Let us call a digraph D cycle-connected if for every pair of vertices u, v ∈ V (D) there exists a cycle containing both u and v. In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge inD, i.e., an edge e ∈ E(D) such that for everyw ∈ V (D) there exists a cycle C such thatw ∈ V (C) and e ∈ E(C)? In his 2001 pap...

In this paper, we study the complete synchronization of Kuramoto model with general network containing a spanning tree, when initial phases are distributed in an open half circle. As lack uniform coercivity digraph, order to capture dissipation structure on network, apply strongly connected graph decomposition criteria Ha et al. [Math. Models Methods Appl. Sci., 30 (2020)] yield hierarchical st...

A digraph is semicomplete if any two vertices are connected by at least one arc and locally the out-neighbourhood in-neighbourhood of vertex induce a digraph. In this paper we study various subclasses digraphs for which give structural decomposition theorems. As consequence obtain several applications, among an answer to conjecture Naserasr first third authors (Aboulker et al., 2021): oriented ...