Ford-Fulkerson algorithm refinement for the cooperation effectiveness increase of intensive and low-density lines

The article describes tools of the railway control on intensive and low-density lines which is directed on the effectiveness increase of low-density line functioning, for the solution of perspective tasks of the railway network functioning and development. For the too realization the oriented graph with the Ford-Fulkerson algorithm which allows determining the maximum flow and the minimum cut for non-oriented graphs. Firstly as values of graph tops inverse values of the station rating and as graph edges inverse values of the railway line class are accepted. The use of this approach allows determining the maximum flow in the system and provides the clear view of relations of transportation capacities of railway lines and stations.

D-Magic Oriented Graphs

In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.

Consecutive Colouring of Oriented Graphs

We consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

On L(h,k)-labelings of oriented graphs

We compare the behaviour of the $L(h,k)$-number of undirected and oriented graphs in terms of maximum degree, highlighting differences between the two contexts. In particular, we prove that, for every $h$ and $k$, oriented graphs with bounded degree in every block of their underlying graph (for instance, oriented trees and oriented cacti) have bounded $L(h,k)$-number, giving an upper bound on this number which is sharp up to a multiplicative factor $4$.