One consequence of an old conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose G is a multigraph with maximum degree ∆, such that no vertex subset S of odd size at most ∆ induces more than (∆+1)(|S|−1)/2 edges. Then G has an edge coloring with ∆ + 1 colors. Here we prove a weakened version of this statement.

The edge strength s(G) of a multigraph G is the minimum number of colors in a minimum sum edge coloring of G. We give closed formulas for the edge strength of bipartite multigraphs and multicycles. These are shown to be classes of multigraphs for which the edge strength is always equal to the chromatic index.

The building of complex, large-scale industrial applications necessitates the integration of models into the problem solving process. This paper describes a modelintegrated program synthesis environment for computerbased system applications. In Model-Integrated Program Synthesis (MIPS), domain-specific, multiple-view models represent the software, its environment and their relationships. Model ...

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted structures and take into account not only the shortest, but also all other connections, which is desirable in many applications. To apply these proximity measures ...

We consider a random generalized railway defined as a random 3-regular multigraph where some vertices are regarded as switches not allowing traffic between any pair of attached edges. It is shown that the probability that the generalized railway is functioning is linear in the proportion of switches. Thus there is no threshold phenomenon for this property.

Diwan and Mubayi asked how many edges of each color could be included in a 3-edge-colored multigraph containing no rainbow triangle. We answer this question under the modest assumption that the multigraphs in question contain at least one edge between every pair of vertices. We also conjecture that this assumption is, in fact, without loss of generality.

Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a “connector” which is a linear combination of circuits with a square-free term. In particular if each circuit of IA with non-square-free terms is balanced, then IA is generated by circuits. As a consequence we prove that the toric id...

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free inte...

The adaptable choosability number of a multigraph G, denoted cha(G), is the smallest integer k such that every edge labeling of G and assignment of lists of size k to the vertices of G permits a list coloring of G in which no edge e = uv has both u and v colored with the label of e. We show that cha grows with ch, i.e. there is a function f tending to infinity such that cha(G) ≥ f(ch(G)).

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