Bang-Jensen, J., P. Hell and G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, Discrete Mathematics 109 (1992) 27-44. Let H be a directed graph whose vertices are called colours. An H-colouring of a digraph G is an assignment of these colours to the vertices of G so that if g is adjacent to g’ in G theq colour(g) is adjacent to colour(g’) in H (i.e., a homomo...

The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot...

In 2020 Bang-Jensen et. al. generalized the Haj\'os join of two graphs to class digraphs and several results for vertex colorings in digraphs. Although, as a consequence these results, digraph can be obtained by constructions (directed identifying non-adjacent vertices), determining obtain is complex problem. particular, et posed problem operations construct symmetric 5-cycle from complete orde...

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.

A noneven digraph is a directed graph with self-loops such that at least one weight-ing of its arcs by 0 or 1 results in no even cycles. A directed cycle in a weighted digraph is even if the sum of weights of its arcs is an even number. The characterisation of 2-connected undirected graphs G such that their symmetric directed counterparts G are noneven, is obtained via a result of Kastelyn conc...

For each pair of positive integers n and k, let G(n,k) denote the digraph whose set of vertices is H = {0,1,2,···, n – 1} and there is a directed edge from a  H to b  H if a  b(mod n). The digraph G(n,k) is symmetric if its connected components can be partitioned into isomorphic pairs. In this paper we obtain all symmetric G (n,k).

The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be real. To overcome this obstacle, several authors have recently defined and studied various Hermitian matrices digraphs or graphs. In work we unify previous offer new perspective on the subject by introducing concept monographs. Moreover, consider questions cospectrality.

Cunningham and Geelen (1997) introduced the notion of independent path-matchings, which generalize both matchings and matroid intersection. Path-matchings are yet generalized to even factors in digraphs by Cunningham and Geelen (2001). Pap (2005) gave a combinatorial algorithm to find a maximum even factor in odd-cycle-symmetric digraphs, in which each arc in any odd dicycle has the reverse arc...