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.
@article{bwmeta1.element.doi-10_7151_dmgt_1973,
author = {Sylvain Gravier and \`Elise Vandomme},
title = {Constant 2-Labellings And An Application To (R, A, B)-Covering Codes},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {891-918},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1973}
}
Sylvain Gravier; Èlise Vandomme. Constant 2-Labellings And An Application To (R, A, B)-Covering Codes. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 891-918. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1973/