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We introduce the concept of constant $2$-labelling of a weighted graph andshow how it can be used to obtain perfect weighted coverings. Roughly speaking,a constant $2$-labelling of a weighted graph is a $2$-colouring of its vertexset which preserves the sum of the weights of black vertices under someautomorphisms. We study this problem on four types of weighted cycles. Ourresults on cycles allow us to determine $(r,a,b)$-codes in $\mathbb{Z}^2$whenever $|a-b|>4$, $r\ge2$ and we give the precise values of $a$ and $b$. Thisis a refinement of Axenovich's theorem proved in 2003.

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