Contrast in Greyscales of Graphs

In this paper,we present the notion of greyscale of a graph, as a colouring of its vertices that uses colours from the continuous spectrum [0, 1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between the colours of its vertices. These edge colours are ordered in lexicographical increasing order and make up the contrast vector of the greyscale. The aim of the contrast problem for the given graph is to find the maximum contrast vector and the greyscales that give rise to it, namely the maximum contrast greyscales. The NP-completeness of this problem is stated by means of a functional relation between the chromatic number and the first component of the maximum contrast vector, named the lightest tone. Thus, we introduce the notion of lightest tone as a new invariant of the graph. The underlying structure of values of maximum contrast greyscales is addressed and we prove that they are linked by rational number sets, which are algorithmically determined. The restricted maximum contrast problem, that is, greyscales with prefixed extreme tones, is also defined and solved in polynomial time for different families of bipartite graphs.