Finite-dimensional T-colorings of graphs

Abstract

This article proposes finite-dimensional T- colorings on simple graphs. The ordinary T-coloring will be the one-dimensional version of our T-colorings. Let d be a positive integer and T be a set of nonnegative integers that contains 0. Denote Z 0 d={(x 1,x 2,…,x d) : x i∈Z 0, 1⩽i⩽d} , where Z 0={0,1,2,…} . A d- dimensional infinite-norm, ||·|| ∞ , is a function from Z 0 d to Z 0 that ||X|| ∞= max{|x i| : 1⩽i⩽d} for X=(x 1,x 2,…,x d) . A d- dimensional one-norm, ||·|| 1 , is a function from Z 0 d to Z 0 that ||X|| 1=∑ i=1 d|x i| for X=(x 1,x 2,…,x d) in Z 0 d . With these norms, we define a d- dimensional T- coloring with respect to p- norm ( p=1 or ∞), f, on a graph G=(V,E) as follows: f:V→Z 0 d such that ||f(u)−f(v)|| p∉T , for all {u,v}∈E. They are denoted by T(d,p)- colorings. Note that both T-colorings for d=1 are same as usual T-colorings. The edge span of a T(d,p)-coloring f on a graph G, esp T(f;d,p) , is the maximum value of {||f(u)−f(v)|| p:{u,v}∈E(G)} . The edge span of G with respect to T(d,p)-coloring is the smallest number m such that there is a T(d,p)-coloring, f with esp T(f;d,p)=m . It is denoted by esp T(G;d,p) . We will show that the edge spans of complete graphs play important roles in seeking edge spans on general graphs. Thus, the goal of this paper is concentrated in finding edge spans of complete graphs. This is same as in the usual T-coloring problem.