Tilings and isoperimetrical shapes. II. Hexagonal lattice
Constant 2-Labellings And An Application To (R, A, B)-Covering Codes
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| >...
Tilings and isoperimetrical shapes I: Square lattice
Hajós' theorem for list colorings of hypergraphs
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 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.
New results about impartial solitaire clobber
Impartial Solitaire Clobber is a one-player version of the combinatorial game Clobber, introduced by Albert in 2002. The initial configuration of Impartial Solitaire Clobber is a graph, such that there is a stone placed on each of its vertex, each stone being black or white. A move of the game consists in picking a stone, and clobbering an adjacent stone of the opposite color. By clobbering we mean that the clobbered stone is removed from the graph, and replaced by the clobbering one. The aim...
Some results on total domination in direct products of graphs
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.
Finding -partitions efficiently
We study the concept of an -partition of the vertex set of a graph , which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph , with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, -part, external...
Finding -partitions efficiently
We study the concept of an -partition of the vertex set of a graph , which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph , with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties: non-list, -part,...
Construction of codes identifying sets of vertices.
A deletion game on graphs: “Le Pic Arête".
Solitaire Clobber played on Hamming graphs.
Adaptive identification in Torii in the King lattice.
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