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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A note on kernels and solutions in digraphs

100%
Harminc M., Soták R.
Discussiones Mathematicae Graph Theory
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1999
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tom 19
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nr 2
237-240
EN
For given nonnegative integers k,s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.
2
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Localization of jumps of the point-distinguishing chromatic index of $K_{n,n}$

100%
Horňák M., Soták R.
Discussiones Mathematicae Graph Theory
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1997
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tom 17
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nr 2
243-251
EN
The point-distinguishing chromatic index of a graph represents the minimum number of colours in its edge colouring such that each vertex is distinguished by the set of colours of edges incident with it. Asymptotic information on jumps of the point-distinguishing chromatic index of $K_{n,n}$ is found.
3
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Generalized Fractional Total Colorings of Graphs

100%
Karafová G., Soták R.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 3
463-473
EN
Let P and Q be additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r/s -fractional (P,Q)-total coloring of a finite graph G = (V,E) is a mapping f, which assigns an s-element subset of the set {1, 2, . . . , r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph with property P, all edges of color i induce a subgraph with property Q and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an r/s -fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number χ″ƒ,P,Q(G) = r/ s . We show in this paper that χ″ƒ,P,Q of a graph G with o(V (G)) vertex orbits and o(E(G)) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of o(V (G)) + o(E(G)) inequalities.
4
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A Note On Vertex Colorings Of Plane Graphs

81%
Fabricia I., Jendrol’ S., Soták R.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 4
849-855
EN
Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V , let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) ≠ c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex coloring with weights in {1, 2, 3}. In a special case, the value 3 is improved to 2.
5
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3-Paths in Graphs with Bounded Average Degree

81%
Jendrol′ S., Maceková M., Montassier M., Soták R.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
339-353
EN
In this paper we study the existence of unavoidable paths on three vertices in sparse graphs. A path uvw on three vertices u, v, and w is of type (i, j, k) if the degree of u (respectively v, w) is at most i (respectively j, k). We prove that every graph with minimum degree at least 2 and average degree strictly less than m contains a path of one of the types [...] Moreover, no parameter of this description can be improved.
6
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Generalized circular colouring of graphs

81%
Mihók P., Oravcová J., Soták R.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 2
345-356
EN
Let P be a graph property and r,s ∈ N, r ≥ s. A strong circular (P,r,s)-colouring of a graph G is an assignment f:V(G) → {0,1,...,r-1}, such that the edges uv ∈ E(G) satisfying |f(u)-f(v)| < s or |f(u)-f(v)| > r - s, induce a subgraph of G with the propery P. In this paper we present some basic results on strong circular (P,r,s)-colourings. We introduce the strong circular P-chromatic number of a graph and we determine the strong circular P-chromatic number of complete graphs for additive and hereditary graph properties.
7
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Generalized Fractional and Circular Total Colorings of Graphs

71%
Kemnitz A., Marangio M., Mihók P., Oravcová J., Soták R.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 3
517-532
EN
Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [ℤr]s be the set of all s-element subsets of ℤr. An (r, s)-fractional (P,Q)-total coloring of G is an assignment h : V (G) ∪ E(G) → [ℤr]s such that for each i ∈ ℤr the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property P, edges with color sets containing color i induce a subgraph of G with property Q, and the color sets of incident vertices and edges are disjoint. If each vertex and edge of G is colored with a set of s consecutive elements of ℤr we obtain an (r, s)-circular (P,Q)-total coloring of G. In this paper we present basic results on (r, s)-fractional/circular (P,Q)-total colorings. We introduce the fractional and circular (P,Q)-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.
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