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1
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Fractional Q-Edge-Coloring of Graphs

100%
Czap J., Mihók P.
Discussiones Mathematicae Graph Theory
|
2013
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tom 33
|
nr 3
509-519
EN
An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let [...] be an additive hereditary property of graphs. A [...] -edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property [...] . In this paper we present some results on fractional [...] -edge-colorings. We determine the fractional [...] -edge chromatic number for matroidal properties of graphs.
2
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Decompositions of Plane Graphs Under Parity Constrains Given by Faces

100%
Czap J., Tuza Z.
Discussiones Mathematicae Graph Theory
|
2013
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tom 33
|
nr 3
521-530
EN
An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f. In this paper we deal with the following question: For which integers k does there exist a facial (facially proper) parity edge coloring of a plane graph G with exactly k colors?
3
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Colouring vertices of plane graphs under restrictions given by faces

100%
Czap J., Jendrol' S.
Discussiones Mathematicae Graph Theory
|
2009
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tom 29
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nr 3
521-543
EN
We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions.
4
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On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

81%
Czap J., Przybyło J., Škrabuľáková E.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
|
nr 1
141-151
EN
A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2n rather than 3n. In particular, we prove that if the size of the smaller partite set is sublinear in n, then |E| = (2 + o(1))n, while the same is not true otherwise.
5
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M2-Edge Colorings Of Cacti And Graph Joins

81%
Czap J., Šugerek P., Ivančo J.
Discussiones Mathematicae Graph Theory
|
2016
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tom 36
|
nr 1
59-69
EN
An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.
6
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On the strong parity chromatic number

81%
Czap J., Jendroľ S., Kardoš F.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 3
587-600
EN
A vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. in [9] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs.
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