T-Span, T-Edge Span Critical Graphs

Given a graph and a finite set of positive integers containing , a -coloring of is a function for all in such that if then . For a -coloring of G, the f-span is the maximum value of over all pairs of vertices of The -span ) is the minimum -span over all -colorings f of . The -edge span of a -coloring is the maximum value of over all edges of . The -edge span ) is the minimum -edge span over all -colorings f of . It is known that ) ) and ) e ) for every graph . In this paper we classify which graphs containing a sub graph such that and ) e ). Also we discuss the Mycielskian of -coloring.