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1
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Equivalent classes for K₃-gluings of wheels

100%
Bielak H.
Discussiones Mathematicae Graph Theory
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1998
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tom 18
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nr 1
73-84
EN
In this paper, the chromaticity of K₃-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K₃-gluings of wheels $W_{q+2}$ and $W_{n-q+2}$ create an χ-equivalent class.
2
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The chromaticity of a family of 2-connected 3-chromatic graphs with five triangles and cyclomatic number six

100%
Bielak H.
Discussiones Mathematicae Graph Theory
|
1998
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tom 18
|
nr 1
99-111
EN
In this note, all chromatic equivalence classes for 2-connected 3-chromatic graphs with five triangles and cyclomatic number six are described. New families of chromatically unique graphs of order n are presented for each n ≥ 8. This is a generalization of a result stated in [5]. Moreover, a proof for the conjecture posed in [5] is given.
3
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A note on the Ramsey number and the planar Ramsey number for C₄ and complete graphs

100%
Bielak H.
Discussiones Mathematicae Graph Theory
|
1999
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tom 19
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nr 2
135-142
EN
We give a lower bound for the Ramsey number and the planar Ramsey number for C₄ and complete graphs. We prove that the Ramsey number for C₄ and K₇ is 21 or 22. Moreover we prove that the planar Ramsey number for C₄ and K₆ is equal to 17.
4
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Multicolor Ramsey numbers for some paths and cycles

100%
Bielak H.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
|
nr 2
209-218
EN
We give the multicolor Ramsey number for some graphs with a path or a cycle in the given sequence, generalizing a results of Faudree and Schelp [4], and Dzido, Kubale and Piwakowski [2,3].
5
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The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

64%
Bielak H., Dąbrowska K.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2015
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tom 69
|
nr 2
EN
The Ramsey number \(R(G, H)\) for a pair of graphs \(G\) and \(H\) is defined as the smallest integer \(n\) such that, for any graph \(F\) on \(n\) vertices, either \(F\) contains \(G\) or \(\overline{F}\) contains \(H\) as a subgraph, where \(\overline{F}\) denotes the complement of \(F\). We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers \(R(K_1+L_n, P_m)\) and \(R(K_1+L_n, C_m)\) for some integers \(m\), \(n\), where \(L_n\) is a linear forest of order \(n\) with at least one edge.
6
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The Turán Number of the Graph 2P5

64%
Bielak H., Kieliszek S.
Discussiones Mathematicae Graph Theory
|
2016
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tom 36
|
nr 3
683-694
EN
We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18.
7
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The density Turan problem for 3-uniform linear hypertrees. An efficient testing algorithm

64%
Bielak H., Powroźnik K.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2018
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tom 72
|
nr 2
EN
Let \(\mathcal{T}=(V,\mathcal{E})\) be a  3-uniform linear hypertree. We consider a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\). We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\) of the hypertree \(\mathcal{T}\), with hyperedge densities satisfying some conditions, such that the hypertree \(\mathcal{T}\) does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree \(\mathcal{T}\) in a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\).
8
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The Turán number of the graph \(3P_4\)

64%
Bielak H., Kieliszek S.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2014
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tom 68
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nr 1
EN
Let \(ex(n, G)\) denote the maximum number of edges in a graph on \(n\) vertices which does not contain \(G\) as a subgraph. Let \(P_i\) denote a path consisting of \(i\) vertices and let \(mP_i\) denote \(m\) disjoint copies of \(P_i\). In this paper we count \(ex(n, 3P_4)\).
9
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Eccentric distance sum index for some classes of connected graphs

64%
Bielak H., Broniszewska K.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2017
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tom 71
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nr 2
EN
In this paper we show some properties of the eccentric distance sum index which is defined as follows \(\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)\). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.
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