Polynomial method for perfect 2-colourings of circulant graphs

Abstract

In this paper we prove that if an infinite circulant graph with k distances has a perfect 2-colouring with parameters (b, c), then b + c 2k + b+c q t for all positive integers t and primes q satisfying b+c gcd(b,c). . .q t .In addition, we show that if b + c = q t , then this necessary condition becomes sufficient for the existence of perfect 2-colourings in circulant graphs.