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1
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Decomposition of Certain Complete Bipartite Graphs into Prisms

100%
Froncek D.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 1
55-62
EN
Häggkvist [6] proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K6n,6n. In [1] Cichacz and Froncek established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into generalized prisms of order 2n. In [2] and [3] Cichacz, Froncek, and Kovar showed decompositions of K3n/2,3n/2 into generalized prisms of order 2n. In this paper we prove that K6n/5,6n/5 is decomposable into prisms of order 2n when n ≡ 0 (mod 50).
2
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Cyclic decompositions of complete graphs into spanning trees

100%
Froncek D.
Discussiones Mathematicae Graph Theory
|
2004
|
tom 24
|
nr 2
345-353
EN
We examine decompositions of complete graphs with an even number of vertices, $K_{2n}$, into n isomorphic spanning trees. While methods of such decompositions into symmetric trees have been known, we develop here a more general method based on a new type of vertex labelling, called flexible q-labelling. This labelling is a generalization of labellings introduced by Rosa and Eldergill.
3
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Rectangular table negotiation problem revisited

64%
Froncek D., Kubesa M.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1114-1120
EN
We solve the last missing case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Suppose we have two negotiating delegations with n=mk members each and we have a seating arrangement such that every day the negotiators sit at m tables with k people of the same delegation at one side of each table. Every person can effectively communicate just with three nearest persons across the table. Our goal is to guarantee that over the course of several days, every member of each delegation can communicate with every member of the other delegation exactly once. We denote by H(k, 3) the graph describing the communication at one table and by mH(k, 3) the graph consisting of m disjoint copies of H(k, 3). We completely characterize all complete bipartite graphs K n,n that can be factorized into factors isomorphic to G =mH(k, 3) for k ≡ 2 (mod 4) by showing that the necessary conditions n=mk and m ≡ 0 mod(3k−2)/4 are also sufficient. This results complement previous characterizations for k ≡ 0, 1, 3 (mod 4) to settle the problem in full.
4
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Problem presented at the workshop in Krynica 2004

64%
Kubesa M., Froncek D.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 2
351
5
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Orthogonal double covers of complete graphs by fat caterpillars

64%
Froncek D., Leck U.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 2
343-349
EN
An orthogonal double cover (ODC) of the complete graph Kₙ by some graph G is a collection of n spanning subgraphs of Kₙ, all isomorphic to G, such that any two of the subgraphs share exactly one edge and every edge of Kₙ is contained in exactly two of the subgraphs. A necessary condition for such an ODC to exist is that G has exactly n-1 edges. We show that for any given positive integer d, almost all caterpillars of diameter d admit an ODC of the corresponding complete graph.
6
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Distance Magic Cartesian Products of Graphs

51%
Cichacz S., Froncek D., Krop E., Raridan C.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 2
299-308
EN
A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ : V → {1, . . . , n} such that the weight of every vertex v, computed as the sum of the labels on the vertices in the open neighborhood of v, is a constant. In this paper, we show that hypercubes with dimension divisible by four are not distance magic. We also provide some positive results by proving necessary and sufficient conditions for the Cartesian product of certain complete multipartite graphs and the cycle on four vertices to be distance magic.
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