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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Vertex-distinguishing edge-colorings of linear forests

100%
Cichacz S., Przybyło J.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 1
95-103
EN
In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) - the irregular coloring number, and hence verify the conjecture when G is a vertex-disjoint union of paths. We also investigate the point-distinguishing chromatic index, χ₀(G), where sets, instead of multisets, are required to be distinct, and determine its value for the same family of graphs.
2
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Rainbow Connection In Sparse Graphs

81%
Kemnitz A., Przybyło J., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
181-192
EN
An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.
3
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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
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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.
4
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Arbitrarily vertex decomposable caterpillars with four or five leaves

71%
Cichacz S., Görlich A., Marczyk A., Przybyło J., Woźniak M.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
291-305
EN
A graph G of order n is called arbitrarily vertex decomposable if for each sequence (a₁,...,aₖ) of positive integers such that a₁+...+aₖ = n there exists a partition (V₁,...,Vₖ) of the vertex set of G such that for each i ∈ {1,...,k}, $V_i$ induces a connected subgraph of G on $a_i$ vertices. D. Barth and H. Fournier showed that if a tree T is arbitrarily vertex decomposable, then T has maximum degree at most 4. In this paper we give a complete characterization of arbitrarily vertex decomposable caterpillars with four leaves. We also describe two families of arbitrarily vertex decomposable trees with maximum degree three or four.
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