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1

Universal third parts of any complete 2-graph and none of DK5

Fortuna A., Skupień Z.

Opuscula Mathematica

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2013

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Vol. 33, no. 4

685--696

EN

It is shown that there is no digraph F which could decompose the complete digraph on 5 vertices minus any 2-arc remainder into three parts isomorphic to F for each choice of the remainder. On the other hand, for each n ≥ 3 there is a universal third part F of the complete 2-graph 2Kn on n vertices, i.e., for each edge subset R of size [formula] mod 3, there is an F-decomposition of 2Kn−R. Using an exhaustive computer-aided search, we find all, exactly six, mutually nonisomorphic universal third parts of the 5-vertex 2-graph. Nevertheless, none of their orientations is a universal third part of the corresponding complete digraph.

2

Zakłócenia w systemie transportu łamanego

Skupień Z.

Logistyka

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2012

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nr 3

PL

Przeplatanie się dwóch (i więcej) gałęzi transportu zwiększa niezawodność całego systemu transportowego. Dla odbiorcy usług przewozowych istotna jest terminowość wykonywanych zadań, a ta wynika pośrednio ze zdatności transportowej infrastruktury, zarówno liniowej jak i punktowej. W referacie przedstawiono niezawodność łamanego systemu transportowego wodno-kolejowego na podstawie danych pozyskanych od zarządców infrastruktury transportu wodnego i kolejowego. Przeanalizowano zdarzenia niepożądane, występujące na drogach wodnych i szlakach kolejowych. Dane te pozwoliły na wyznaczenie gotowości poszczególnych gałęzi i systemu transportu łamanego.

EN

Interweaving of the two (or more) means of transport increases the reliability of the entire transport system. For the customers of transportation services it is essential to have tasks done on time, and this result indirectly from the airworthiness of transport infrastructure, both linear and point. The paper presents the reliability of broken water-rail transport system on the basis of data obtained from water transport infrastructure managers and railway stations. Adverse events have been analyzed occurring in the waterways and rail routes. These data allowed to determine readiness of individual branches and broken transportation system.

3

Edge decompositions of multigraphs into multi-2-paths

Kratochvil J., Lonc Z., Meszka M., Skupień Z.

Opuscula Mathematica

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2004

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Vol. 24, no. 1

97-102

EN

We establish the computational time complexity of the existence problem of a decomposition of an instance multigraph into isomorphic 3-vertex paths with multiple edges. If the two edge multiplicities are distinct, the problem is NPC; if mutually equal then polynomial.

4

Twierdzenie Cantora-Bernsteina : dowody znane-nieznane

Skupień Z.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

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2003

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[Z] 39

85-94

5

Large minimal irregular digraphs

Dziechcińska-Halamoda Z., Majcher Z., Michael J., Skupień Z.

Opuscula Mathematica

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2003

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Vol. 23

21-24

EN

A digraph is called irregular if its distinct vertices have distinct degree pairs. An irregular digraph is called minimal if the removal of any arc results in a non-irregular digraph. A large minimal irregular digraph Fn of order n is constructed if n is the sum of initial positive integers. It is easily seen that the minimum and maximum sizes among n-vertex irregular digraphs are asymptotic to [formula] and n2, respectively. It appears that the size of Fn is asymptotic to n2, too. Similarly, a minimal irregular oriented graph Hn is constructed such that the size of Hn is asymptotic to 1/2n2 whence it is asymptotically the largest size among n-vertex oriented graphs whether irregular or not.

6

Prosty dowód twierdzenia Cantora-Bernsteina

Skupień Z.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

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1999

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[Z] 35

49-53