T-Coloring on Folded Hypercubes

Abstract

Given a graph G = (V, E) and a set T of non-negative integers containing 0, a T -coloring of G is an integer function f of the vertices of G such that |f(u) − f(v)| / ∈ T whenever uv ∈ E. The edge-span of a Tcoloring f is the maximum value of |f(u) − f(v)| over all edges uv, and the T -edge-span of a graph G is the minimum value of the edge-span among all possible T -colorings of G. This paper discusses the T -edge span of the folded hypercube network of dimension n for the k-multiple-of-

T-colorings, divisibility and circular chromatic number

“…where esp(c) = max{|c(u) − c(v)| : uv ∈ E} is the edge span of c (if G is an empty graph then esp(c) = 0). If we replace esp(c) by sp(c) (the span of c, i.e., max{|c(u) − c(v)| : u, v ∈ V }) we will receive the T -span of G. Both parameters were studied by many authors, there are results concerning computational complexity of the problem of computing sp T (G) [2,3], the behaviour of the greedy algorithm [7] and formulas describing sp T (G) and esp T (G) for some T -sets T and some graphs G [8,9,13]. The remainder of the paper is organized as follows.…”