Opuscula Mathematica
Opuscula Math. 37, no. 4 (2017), 535-556

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

Ewa Drgas-Burchardt
Agata Drzystek
Abstract. 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 \(D_1\)-sum-choice-number of \(G\). In the paper we investigate the \(D_1\)-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the \(D_1\)-sum-choice-number for each grid \(P_n\square P_m\), when at least one of the numbers \(n\), \(m\) is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the \(D_1\)-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs.
Keywords: sum-list colouring, acyclic colouring, grids, generalized Petersen graphs.
Mathematics Subject Classification: 05C30, 05C15.
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About the authors ▼
Ewa Drgas-Burchardt
University of Zielona Góra, Faculty of Mathematics, Computer Science and Econometrics, ul. Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
Agata Drzystek
University of Zielona Góra, Faculty of Mathematics, Computer Science and Econometrics, ul. Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
About the article ▼
Communicated by Ingo Schiermeyer.
Received: 2016-08-31.
Revised: 2016-12-23.
Accepted: 2017-01-11.
Published online: 2017-04-28.
Cite this article as ▼
Ewa Drgas-Burchardt, Agata Drzystek, Acyclic sum-list-colouring of grids and other classes of graphs, Opuscula Math. 37, no. 4 (2017), 535-556, http://dx.doi.org/10.7494/OpMath.2017.37.4.535.
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