Polynomial method for perfect 2-colourings of circulant graphs

TL;DR

This paper establishes a polynomial-based criterion for perfect 2-colourings in circulant graphs, linking parameters to algebraic conditions, and identifies when these conditions are both necessary and sufficient.

Contribution

It introduces a polynomial method to characterize perfect 2-colourings in circulant graphs, providing new necessary and sufficient conditions based on algebraic divisibility.

Findings

Derived a bound on parameters for perfect 2-colourings.

Proved sufficiency of conditions when sum of parameters equals a prime power.

Connected algebraic divisibility conditions with graph colouring existence.

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 $b + c \leq 2k + \frac{b+c}{q^t}$ for all positive integers $t$ and primes $q$ satisfying $\frac{b+c}{gcd(b,c)}\vdots q^t$. 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.