Graph Imperfection

We are interested in colouring a graph G=(V, E) together with an integral weight or demand vector x=(xv: v∈V) in such a way that xv colours are assigned to each node v, adjacent nodes are coloured with disjoint sets of colours, and we use as few colours as possible. Such problems arise in the design of cellular communication systems, when radio channels must be assigned to transmitters to satisfy demand and avoid interference. We are particularly interested in the ratio of chromatic number to clique number when some weights are large. We introduce a relevant new graph invariant, the “imperfection ratio” imp(G) of a graph G, present alternative equivalent descriptions, and show some basic properties. For example, imp(G)=1 if and only if G is perfect, imp(G)=imp(G) where G denotes the complement of G, and imp(G)=g/(g−1) for any line graph G where g is the minimum length of an odd hole (assuming there is an odd hole).