Constant 2-labellings and an application to (r, a, b)-covering codes

We introduce the concept of constant 2-labelling of a vertex-weighted graph and show how it can be used to obtain perfect weighted coverings. Roughly speaking, a constant 2-labelling of a vertex-weighted graph is a black and white colouring of its vertex set which preserves the sum of the weights of black vertices under some automorphisms. We study constant 2-labellings on four types of vertex-weighted cycles. Our results on cycles allow us to determine (r, a, b)-codes in Z whenever |a − b| > 4, r ≥ 2 and we give the precise values of a and b. This is a refinement of Axenovich’s theorem proved in 2003.