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On (2-d)-kernels in the cartesian product of graphs

100%
Bednarz P., Włoch I.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2016
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tom 70
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nr 2
EN
In this paper we study the problem of the existence of (2-d)-kernels in the cartesian product of graphs. We give sufficient conditions for the existence of (2-d)-kernels in the cartesian product and also we consider the number of (2-d)-kernels.
2
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Grundy number of graphs

100%
Effantin B., Kheddouci H.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
|
nr 1
5-18
EN
The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 ≤ i ≤ k, is adjacent to (i-1) vertices colored with each color j, 1 ≤ j ≤ i -1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an algorithm to generate all graphs for a given Grundy number.
3
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Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph

100%
Mojdeh D.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
59-72
EN
In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cₘ × Kₙ, c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number $d(K₃×C_{2r},4)$, which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.
4
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Categories of functors between categories with partial morphisms

63%
Vogel H.-J.
Discussiones Mathematicae - General Algebra and Applications
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2005
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tom 25
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nr 1
39-87
EN
It is well-known that the composition of two functors between categories yields a functor again, whenever it exists. The same is true for functors which preserve in a certain sense the structure of symmetric monoidal categories. Considering small symmetric monoidal categories with an additional structure as objects and the structure preserving functors between them as morphisms one obtains different kinds of functor categories, which are even dt-symmetric categories.
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