Graph imperfection and channel assignment

Given a graph G and a non-negative integer weight or demand xv for each node v, a colouring of the pair (G, x) is an assignment to each node v of a set of xv colours such that adjacent nodes receive disjoint sets of colours. We are interested in such colourings when the maximum demand xv is large. An application where such colouring problems arise is in the design of cellular radio communication systems: we need to assign sets of radio frequency channels (colours) to transmitters (nodes), using few channels but avoiding excessive interference. We introduce here a new relevant graph invariant, the ‘imperfection ratio’ imp(G), and present some initial investigations concerning it. This invariant measures how well clique-based lower bounds approximate the number of colours required, when there are large demands. It satisfies for example imp(G) ≥ 1, imp(G) = 1 if and only if G is perfect, imp(G) = imp(Ḡ) where Ḡ denotes the complement of G, and if G is a line graph then imp(G) = g g−1 where g is the minimum length of an odd hole (if there is one).