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Packing of three copies of a digraph into the transitive tournament

100%
Pilśniak M.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 3
443-456
EN
In this paper, we show that if the number of arcs in an oriented graph G (of order n) without directed cycles is sufficiently small (not greater than [2/3] n-1), then there exist arc disjoint embeddings of three copies of G into the transitive tournament TTₙ. It is the best possible bound.
2
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A note on packing of two copies of a hypergraph

64%
Pilśniak M., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
45-49
EN
A 2-packing of a hypergraph 𝓗 is a permutation σ on V(𝓗) such that if an edge e belongs to 𝓔(𝓗), then σ (e) does not belong to 𝓔(𝓗). We prove that a hypergraph which does not contain neither empty edge ∅ nor complete edge V(𝓗) and has at most 1/2n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1/2n edges shows that this result cannot be improved by increasing the size of 𝓗.
3
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On cyclically embeddable (n,n)-graphs

51%
Görlich A., Pilśniak M., Woźniak M.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
85-104
EN
An embedding of a simple graph G into its complement G̅ is a permutation σ on V(G) such that if an edge xy belongs to E(G), then σ(x)σ(y) does not belong to E(G). In this note we consider the embeddable (n,n)-graphs. We prove that with few exceptions the corresponding permutation may be chosen as cyclic one.
4
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Structural Properties of Recursively Partitionable Graphs with Connectivity 2

51%
Baudon O., Bensmail J., Foucaud F., Pilśniak M.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
89-115
EN
A connected graph G is said to be arbitrarily partitionable (AP for short) if for every partition (n1, . . . , np) of |V (G)| there exists a partition (V1, . . . , Vp) of V (G) such that each Vi induces a connected subgraph of G on ni vertices. Some stronger versions of this property were introduced, namely the ones of being online arbitrarily partitionable and recursively arbitrarily partitionable (OL-AP and R-AP for short, respectively), in which the subgraphs induced by a partition of G must not only be connected but also fulfil additional conditions. In this paper, we point out some structural properties of OL-AP and R-AP graphs with connectivity 2. In particular, we show that deleting a cut pair of these graphs results in a graph with a bounded number of components, some of whom have a small number of vertices. We obtain these results by studying a simple class of 2-connected graphs called balloons.
5
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Dense Arbitrarily Partitionable Graphs

51%
Kalinowski R., Pilśniak M., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 1
5-22
EN
A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with [...] ‖G‖ > (n−42)+12 $||G||\; > \;\left( {\matrix{{n - 4} \cr 2 \cr } } \right) + 12$ edges is AP or belongs to few classes of exceptional graphs.
6
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Distinguishing Cartesian Products of Countable Graphs

45%
Estaji E., Imrich W., Kalinowski R., Pilśniak M., Tucker T.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
155-164
EN
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
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