An overview of satellite communication scheduling and its relation to edge-coloring of multigraphs is given. Then a theorem about a restricted class of multigraphs is proved to obtain conditions for scheduling in a satellite communications network of practical interest Some limitations of the multigraph model are then discussed.

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means no order at least two is, in this way, irregular. However, multigraph can be Chartrand et al., 1988, posed following problem: loopless multigraph, how one determine fewest parallel edges required to ensure all degrees? problem known as labeling and, for its solution, al. introduced irregu...

We give a simpler proof of Seymour’s Theorem on edge-coloring series-parallel multigraphs and derive a linear-time algorithm to check whether a given series-parallel multigraph can be colored with a given number of colors.

Let Kn denote the complete 2-fold multigraph of order n and let G be a bipartite subgraph of K4. We find necessary and sufficient conditions for the existence of a G-decomposition of Kn.

The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i 6= j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extende...

We give a complete solution to the existence problem for complementary P3-decompositions of the complete multigraph, where P 3 denotes the path of length 3. A complementary decomposition 2>.K v (P 3 , P 3) is an edge decomposition of the complete multigraph >'Kv into P 3 '8 with the property that upon taking the complement of each path one obtains a second decomposition of >'K3 into P3 '8 (wher...

An n-multigraph G = (V ;Ei | i ∈ I) is a complete graph G = (V,E) whose edges are covered by n = |I| sets E = ∪i∈IEi, some of which might be empty. If this cover is a partition, then we call G an n-graph. We say that an n-graph G′ = (V ;E′ i | i ∈ I) is an edge-subgraph of an n-multigraph G = (V ;Ei | i ∈ I) if E′ i ⊆ Ei for all i ∈ I. We denote by ∆ the n-graph on three vertices with three non...

We present a construction that allows us to define limit object of Banach space decorated graph sequences in generalized homomorphism density sense. This general functional analytic framework provides universal language for various combinatorial notions. In particular it makes possible assign objects multigraph are convergent the sense node-and-edge numbers, and generalizes theory with compact ...