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

100%
Gravier S., Vandomme È.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
nr 4
891-918
EN
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.
2
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Hajós' theorem for list colorings of hypergraphs

81%
Benzaken C., Gravier S., Skrekovski R.
Discussiones Mathematicae Graph Theory
|
2003
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tom 23
|
nr 2
207-213
EN
A well-known theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph $K_{k+1}$ by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to list-colorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós' theorem to list-colorings of hypergraphs.
3
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Some results on total domination in direct products of graphs

81%
Dorbec P., Gravier S., Klavžar S., Spacapan S.
Discussiones Mathematicae Graph Theory
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2006
|
tom 26
|
nr 1
103-112
EN
Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.
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