T-Coloring of Certain Networks

Given a graph G and a finite set T of non-negative integers containing zero, a T -coloring of G is a non-negative integer function f defined on V ( G ) such that | f ( x ) - f ( y ) | ∉ T whenever ( x , y ) ∈ E ( G ) . The span of T -coloring is the difference between the largest and smallest colors, and the T -span of G is the minimum span over all T -colorings f of G . The edge span of a T -coloring is the maximum value of | f ( x ) - f ( y ) | over all edges ( x , y ) ∈ E ( G ) , and the T -edge span of G is the minimum edge span over all T -colorings f of G . In this paper, we compute T -span and T -edge span of crown graph, circular ladder and mobius ladder, generalized theta graph, series-parallel graph and wrapped butterfly network.