Circular colouring and graph homomorphism

Abstract

For any pair of integers p, q such that (p, q) = 1 and p ≥ 2q, the graph Gqp has vertices {0, 1, ... , p - 1} and edges {ij : q ≤ |i - j| ≤ p - q}. These graphs play the same role in the study of circular chromatic number as that played by the complete graphs in the study of chromatic number. The graphs Gqp share many properties of the complete graphs. However, there are also striking differences between the graphs Gqp and the complete graphs. We shall prove in this paper that for many pairs of integers p, q, one may delete most of the edges of Gqp so that the resulting graph still has circular chromatic number p/q. To be precise, we shall prove that for any number r > 2, there exists a rational number p/q (where (p, q) = 1) which is less than r but arbitrarily close to r, such that Gqp contains a subgraph H with χc(H) = χc(Gqp) = p/q and |E(H)| = O(√|E(Gqp)|). This is in sharp contrast to the fact that the complete graphs are edge critical, that is, the deletion of any edge will decrease its chromatic number and its circular chromatic number.