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Corrigendum to "acyclic sum-list-colouring of grids and other classes of graphs" [Opuscula Math. 37, no. 4 (2017), 535 556]

Drgas-Burchardt E., Drzystek A.

Opuscula Mathematica

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2018

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Vol. 38, no. 6

899--901

EN

This note provides some minor corrections to the article [Acyclic sum-list-colouring of grids and other classes of graphs, Opuscula Math. 37, no. 4 (2017), 535-556].

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Acyclic sum-list-colouring of grids and other classes of graphs

Drgas-Burchardt E., Drzystek A.

Opuscula Mathematica

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2017

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Vol. 37, no. 4

535--556

EN

In this paper we consider list colouring of a graph G in which the sizes of lists assigned to different vertices can be different. We colour G from the lists in such a way that each colour class induces an acyclic graph. The aim is to find the smallest possible sum of all the list sizes, such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes. This invariant is called the D1-sum-choice-number of G. In the paper we investigate the D1-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the D1-sum-choice-number for each grid [formula], when at least one of the numbers n, rn is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the D1-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs.

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The number of stable matchings in models of the Gale-Shapley type with preferences given by partial orders

Drgas-Burchardt E.

Operations Research and Decisions

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2015

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Vol. 25, No. 1

5--15

EN

From the famous Gale–Shapley theorem we know that each classical marriage problem admits at least one stable matching. This fact has inspired researchers to search for the maximum number of possible stable matchings, which is equivalent to finding the minimum number of unstable matchings among all such problems of size n. In this paper, we deal with this issue for the Gale–Shapley model with preferences represented by arbitrary partial orders. Also, we discuss this model in the context of the classical Gale–Shapley model.

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A note on a list colouring of hypergraphs

Drgas-Burchardt E.

Opuscula Mathematica

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2004

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Vol. 24, no. 2

171-175

EN

In the note we present two results. The first of them gives a sufficient condition for a colouring of a hypergraph from an assigned list. It generalises the analogous fact for graphs. The second result states that for every k ≥ 3 and every l ≥ 2, a distance between the list chromatic number and the chromatic number can be arbitrarily large in the class of k-uniform hypergraphs with the chromatic number bounded below by l. A similar result for k-uniform, 2-colorable hypergraphs is known but the proof techniques are different.