Span of T-Colorings Multigraphs

Abstract

Suppose G is a graph and T is a set of nonnegative integers. A T-coloring of G is an assignment of a positive integer f(x) to each vertex x of G so that if x and y are joined by an edge of G, then |f (x) – f (y)| is not in T. Here ,the vertices of G are transmitters, an edge represents interference, f(x) is a television or radio channel assigned to x, and T is a set of disallowed separations for channels assigned to interfering transmitters. The span of a T-coloring of G equals max |f (x) – f (y)| , where the maximum is taken over all edges {x,y} ∈ E(G) . The minimum order, and minimum span, where the minimum is taken over all T-colorings of G, are denoted by ( ) T 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.