Constant 2-Labellings And An Application To (R, A, B)-Covering Codes

• Main Authors: Gravier Sylvain, Vandomme Èlise
• Format: Article
• Language: English
• Published: Sciendo 2017-11-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: covering codes; weighted codes; infinite grid; vertex-weighted graphs.
• Online Access: https://doi.org/10.7151/dmgt.1973

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 Z2 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.