Graph Imperfection with a Co-Site Constraint

We are interested in a version of graph colouring where there is a ‘co-site’ constraint value k. Given a graph G with a non-negative integral demand xv at each node v, we must assign xv positive integers (colours) to each node v such that the same integer is never assigned to adjacent nodes, and two distinct integers assigned to a single node differ by at least k. The aim is to minimise the span, that is the largest integer assigned to a node. This problem is motivated by radio channel assignment where one has to assign frequencies to transmitters so as to avoid interference. We compare the span with a clique-based lower bound when some of the demands are large. We introduce the relevant graph invariant, the k-imperfection ratio, give equivalent definitions and investigate some of its properties. The k-imperfection ratio is always at least 1: we call a graph k-perfect when it equals 1. Then 1-perfect is the same as perfect; and we see that for many classes of perfect graphs, each graph in the class is k-perfect for all k. These classes include comparability graphs, co-comparability graphs and line-graphs of bipartite graphs.