Multiscale resolution database MRDB include not only several levels of details (ie preciseness), but also several layers of peculiarity. The data preciseness means the variable levels of details for the fixed set of attributes. Less details leads to data aggregation, more details leads to data de-aggregation. The data peculiarity means the growing and reduction of the number of variable attribu...

Let G = (V (G), E(G)) be a multigraph. The degree of a vertex x in G is denoted by d(x), the greatest degree of a vertex – by ∆(G), the chromatic index of G – by χ ′ (G). Let R ⊆ V (G). An interval (respectively, continuous) on R t-coloring of a multigraph G is a proper coloring of edges of G with the colors 1, 2,. .. , t, in which each color is used at least for one edge, and the edges inciden...

A linear-time algorithm for decomposing a graph into split components is presented. The algorithm uses a new graph transformation technique to gradually transform the given graph so that every split component in it is transformed into a subgraph with very simple structure which can be easily identified. Once the split components are determined, the triconnected components of the graph are easil...

In path factorization, H. Wang [1] gives the necessary and sufficient conditions for the existence of P_k-factorization of a complete bipartite graph for k, an even integer. Further, Beiling Du [2] extended the work of H. Wang, and studied the P_2k-factorization of complete bipartite multigraph. For odd value of k the work on factorization was done by a number of researchers. P_3-factorization ...

A path or cycle in an edge-coloured multigraph is called alternating if its successive edges differ in colour. We survey results of both theoretical and algorithmic character concerning alternating cycles and paths in edge-coloured multigraphs. We also show useful connections between the theory of paths and cycles in bipartite digraphs and the theory of alternating paths and cycles in edge-colo...

Graphs with the above property seem to behave well also with respect to other, more complicated, expansion-type properties. Indeed, the author was motivated to study this question by a paper communicated to him in May 2002 (the latest version of the paper is available from URL http://www-math.mit.edu/∼vempala/papers/tspinapprox.ps). In this paper, Papadimitriou and Vempala established approxima...

We consider cycles and paths in multigraphic realizations of a degree sequence Q. in particular we show that there exists a realization of d in which no cycle has order greater than three and no path has length greater than four. In addition we show which orders of cycles and which lengths of paths exist in some realization of d.

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies. This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Euroco...

We prove that the vertex set of any twin-free loopless multigraph G has an embedding into some point set P of some Euclidean space R , such that the automorphism group of G is isomorphic to the isometry group of R globally preserving P .

Let G be a multigraph with a fixed orientation D and let Γ be a group. Let F(G, Γ ) denote the set of all functions f : E(G) → Γ . A multigraph G is Γ -colorable if and only if for every f ∈ F(G, Γ ), there exists a Γ -coloring c : V (G) → Γ such that for every e = uv ∈ E(G) (assumed to be directed from u to v), c(u)c(v)−1 ≠ f (e). We define the group chromatic number χg (G) to be the minimum i...