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A note on face coloring entire weightings of plane graphs

100%
Jendrol S., Šugerek P.
Discussiones Mathematicae Graph Theory
|
2014
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tom 34
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nr 2
421-426
EN
Given a weighting of all elements of a 2-connected plane graph G = (V,E, F), let f(α) denote the sum of the weights of the edges and vertices incident with the face _ and also the weight of _. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and _ sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3
2
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Light edges in 1-planar graphs with prescribed minimum degree

100%
Hudák D., Šugerek P.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
|
nr 3
545-556
EN
A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains just the selected edge type).
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M2-Edge Colorings Of Cacti And Graph Joins

81%
Czap J., Šugerek P., Ivančo J.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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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.
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