Strong T-Coloring of Graphs

Abstract

A -coloring of a graph is the generalized coloring of a graph. Given a graph and a finite set T of positive integers containing , a -coloring of is a function for all in such that if then We define Strong -coloring (S -coloring , in short), as a generalization of -coloring as follows. Given a graph and a finite set of positive integers containing , a S -coloring of is a function for all in such that if then and for any two distinct edges in . The S -Chromatic number of is the minimum number of colors needed for a S -coloring of and it is denoted by For a S coloring of a graph we define the -span is the maximum value of over all pairs of vertices of and the S -span is defined by = min where the minimum is taken over all ST-coloring c of G. Similarly the -edgespan is the maximum value of over all edges of and the S -edge span is defined by = min where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S -chromatic number, , and of graphs.