An edge-coloring of a multigraph with colors 1, is called an interval coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that if is a connected cubic multigraph (a connected cubic graph) that admits an interval coloring, then G 2, , t ... t −

The maximum of the maximum degree and the 'odd set quotients' provides a well-known lower bound 4)(G) for the chromatic index of a multigraph G. Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals qS(G) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed qS(G) + 1. We identify those multigraphs G of order 9 and 10 whose chromat...

It is proved that a connected multigraph G which is the union of two edge-disjoint paths has another decomposition into two paths with the same set, U , of endvertices provided that the multigraph is neither a path nor cycle. Moreover, then the number of such decompositions is proved to be even unless the number is three, which occurs exactly if G is a tree homeomorphic with graph of either sym...

Graph based models of hierarchical systems are usually seen as ‘graph equipped with some refinements’, understood as the (homo)morphisms or (bi)simulations. In such a model it is not possible to consider phenomena happened on different levels of the system. We propose a new formalism of directed multi-graph allowing to see a hierarchical system similar as a formula of second order logic, i.e. t...

The paper discusses a comprehensive, model-based approach for the design and implementation of intelligent controllers. The system has been implemented in the framework of the Multigraph Architecture. The Multigraph Architecture is a layered system, which includes a parallel, graph computation model, the corresponding execution environment, and software tools supporting the interactive, graphic...

1 Edge-disjoint decompositions Some of the graphs in this section are allowed to have multiple edges. They will be referred to as multigraphs. All graphs are assumed to have vertex set [n] = {1, 2, . . . , n}. Multiple edges between vertices of a multigraph G are regarded as distinct members of its edge set, E(G). We say that a collection of multigraphs1 Gi, i ∈ [m] is an edge-disjoint decompos...