T-Coloring of product graphs

Abstract

Given a graph [Formula: see text] and a finite set [Formula: see text] of positive integers containing [Formula: see text] a [Formula: see text]-coloring of [Formula: see text] is a function [Formula: see text] for all [Formula: see text] in [Formula: see text] such that if [Formula: see text] then [Formula: see text] For a [Formula: see text]-coloring [Formula: see text] of [Formula: see text], the [Formula: see text]-span [Formula: see text] is the maximum value of [Formula: see text] over all pairs [Formula: see text] of vertices of [Formula: see text] The [Formula: see text]-span [Formula: see text] is the minimum [Formula: see text]-span overall [Formula: see text]-colorings of [Formula: see text]. The [Formula: see text]-edge span of a [Formula: see text]-coloring [Formula: see text] is the maximum value of [Formula: see text] over all edges [Formula: see text] of [Formula: see text]. The [Formula: see text]-edge span [Formula: see text] is the minimum [Formula: see text]-edge span overall [Formula: see text]-colorings of [Formula: see text]. This paper discusses the [Formula: see text]-span and [Formula: see text]-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.