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1
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A note on uniquely embeddable graphs

100%
Woźniak M.
Discussiones Mathematicae Graph Theory
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1998
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tom 18
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nr 1
15-21
EN
Let G be a simple graph of order n and size e(G). It is well known that if e(G) ≤ n-2, then there is an embedding G into its complement [G̅]. In this note, we consider a problem concerning the uniqueness of such an embedding.
2
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On cyclically embeddable graphs

100%
Woźniak M.
Discussiones Mathematicae Graph Theory
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1999
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tom 19
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nr 2
241-248
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 some families of embeddable graphs such that the corresponding permutation is cyclic.
3
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A Note on Uniquely Embeddable Forests

64%
Otfinowska J., Woźniak M.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
193-201
EN
Let F be a forest of order n. It is well known that if F 6= Sn, a star of order n, then there exists an embedding of F into its complement F. In this note we consider a problem concerning the uniqueness of such an embedding.
4
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New sufficient conditions for hamiltonian and pancyclic graphs

64%
Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
29-38
EN
For a graph G of order n we consider the unique partition of its vertex set V(G) = A ∪ B with A = {v ∈ V(G): d(v) ≥ n/2} and B = {v ∈ V(G):d(v) < n/2}. Imposing conditions on the vertices of the set B we obtain new sufficient conditions for hamiltonian and pancyclic graphs.
5
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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.
6
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A note on a new condition implying pancyclism

52%
Flandrin E., Li H., Marczyk A., Woźniak M.
Discussiones Mathematicae Graph Theory
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2001
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tom 21
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nr 1
137-143
EN
We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n/2,n/2}$.
7
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Rainbow Connection In Sparse Graphs

52%
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.
8
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On cyclically embeddable (n,n)-graphs

52%
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.
9
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Dense Arbitrarily Partitionable Graphs

52%
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.
10
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A note on maximal common subgraphs of the Dirac's family of graphs

52%
Bucko J., Mihók P., Saclé J.-F., Woźniak M.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
385-390
EN
Let 𝓕 ⁿ be a given set of unlabeled simple graphs of order n. A maximal common subgraph of the graphs of the set 𝓕 ⁿ is a common subgraph F of order n of each member of 𝓕 ⁿ, that is not properly contained in any larger common subgraph of each member of 𝓕 ⁿ. By well-known Dirac's Theorem, the Dirac's family 𝓓𝓕 ⁿ of the graphs of order n and minimum degree δ ≥ [n/2] has a maximal common subgraph containing Cₙ. In this note we study the problem of determining all maximal common subgraphs of the Dirac's family $𝓓 𝓕 ^{2n}$ for n ≥ 2.
11
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A Note on Neighbor Expanded Sum Distinguishing Index

45%
Flandrin E., Li H., Marczyk A., Saclé J.-F., Woźniak M.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
29-37
EN
A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.
12
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Chvátal-Erdos condition and pancyclism

45%
Flandrin E., Li H., Marczyk A., Schiermeyer I., Woźniak M.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
335-342
EN
The well-known Chvátal-Erdős theorem states that if the stability number α of a graph G is not greater than its connectivity then G is hamiltonian. In 1974 Erdős showed that if, additionally, the order of the graph is sufficiently large with respect to α, then G is pancyclic. His proof is based on the properties of cycle-complete graph Ramsey numbers. In this paper we show that a similar result can be easily proved by applying only classical Ramsey numbers.
13
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Arbitrarily vertex decomposable caterpillars with four or five leaves

45%
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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