T-colorings of graphs

Abstract

Given a finite set T of positive integers containing {0};, a T-coloring of a simple graph G is a nonnegative integer function f defined on the vertex set of G, such that if { u, v}; ϵ E( G) then ¦ f( u) - f( v)¦ ∉ T. The T-span of a T-coloring is defined as the difference of the largest and smallest colors used; the T-span of G, sp T ( G), is the minimum span over all T-colorings of G. It is known that the T-span of G satisfies sp T ( K ω( G) ) ⩽ sp T ( G) ⩽ sp T ( K x( G) ). When T is an r-initial set (Cozzens and Roberts, 1982), or a k multiple of s set (A. Raychaudhuri, 1985), then sp T ( G) = sp T ( K x( G) ) for all graphs G. Using graph homomorphisms and a special family of graphs, we characterize those T's with equality sp T ( G) =sp T ( K x( G) ) for all graphs G. We discover new T's with the same result. Furthermore, we get a necessary and sufficient condition of equality sp T ( G) = sp T ( K m ) for all graphs G with X ( G) = m.