T-Colorings of Multigraphs

T-colorings of multigraphs are a more practical case of T-colorings of graphs where interference may occur on different level. A multigraph G can be partitioned into K distinct sets, so we represent G as G(V,G₀,G₁,G₂,...,G_K-1). The span of a T-coloring of G equals max |f(x)-f(y)|, where the maximum is taken over all edges {x, y} ϵ E(G). The minimum order, and minimum span, where the minimum is taken over all T-colorings of G, are denoted by X_T(G), and sp_T(G), respectively. We will show several previous results of multigraphs, and we also will present a new algorithm to compute sp_T(G) of multigraphs.