T-Coloring of product graphs.

Given a graph G = (V , E) and a finite set T of positive integers containing 0 , a T -coloring of G is a function f : V (G) → Z + ∪ { 0 } for all u ≠ w in V (G) such that if u w ∈ E (G) then | f (u) − f (w) | ∉ T. For a T -coloring f of G , the f -span sp T f (G) is the maximum value of | f (u) − f (w) | over all pairs u , w of vertices of G. The T -span sp T (G) is the minimum f -span overall T -colorings of G. The f -edge span of a T -coloring esp T f (G) is the maximum value of | f (u) − f (w) | over all edges u w of G. The T -edge span esp T (G) is the minimum f -edge span overall T -colorings of G. This paper discusses the T -span and T -edge span of Cartesian, Join, Union and Tensor products of graphs. Also, we discuss the relation between Restricted span and Restricted edge span of graphs. [ABSTRACT FROM AUTHOR]