Geodesic
Constant 2-Labellings And An Application To (R, A, B)-Covering Codes
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 891-918

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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^2 whenever |a − b| gt; 4, r ≥ 2 and we give the precise values of a and b. This is a refinement of Axenovich’s theorem proved in 2003.
Keywords: covering codes, weighted codes, infinite grid, vertex-weighted graphs 
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Gravier, Sylvain; Vandomme, Èlise. Constant 2-Labellings And An Application To (R, A, B)-Covering Codes. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 891-918. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a3/