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

Opuscula Mathematica

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.

Referred to by Corrigendum to "Acyclic sum-list-colouring of grids and other classes of graphs"

Article: Opuscula Math. 38, no. 6 (2018), 899-901, https://doi.org/10.7494/OpMath.2018.38.6.899

Keywords: sum-list colouring, acyclic colouring, grids, generalized Petersen graphs.

Mathematics Subject Classification: 05C30, 05C15.

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  1. L.W. Beineke, R.C. Vandell, Decycling graphs, J. Graph Theory 25 (1996), 59-77.
  2. M. Borowiecki, P. Mihók, Hereditary properties of graphs, [in:] V.R. Kulli (ed.), Advances in Graph Theory, Vishawa International Publication, Gulbarga, 1991, 41-68.
  3. R. Diestel, Graph Theory, 2nd ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, New York, 2000.
  4. E. Drgas-Burchardt, A. Drzystek, General and acyclic sum-list-colouring of graphs, Appl. Anal. Discrete Math. 10 (2016) 2, 479-500.
  5. L. Gao, X. Xu, J. Wang, D.Zhu, Y. Yang, The decycling number of generalized Petersen graphs, Discrete Appl. Math. 181 (2015), 297-300.
  6. P. Erdös, A.L. Rubin, H. Taylor, Choosability in graphs, [in:] Proceedings West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata CA, Congr. Numer. 26 (1979).
  7. G. Isaak, Sum list coloring \(2 \times n\) arrays, Electron. J. Combin. 9 (2002) #N8.
  8. E. Kubicka, A.J. Schwenk, An introduction to chromatic sums, [in:] Proceedings of the Seventeenth Annual ACM Computer Sciences Conference, ACM Press (1989), 39-45.
  9. M. Lastrina, List-coloring and sum-list-coloring problems on graphs, Ph.D. Thesis, Iowa State University, 2012.
  10. F.L. Luccio, Almost exact minimum feedback vertex sets in meshes and butterflies, Inform. Process. Lett. 66 (1998), 59-64.
  • 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
  • Communicated by Ingo Schiermeyer.
  • Received: 2016-08-31.
  • Revised: 2016-12-23.
  • Accepted: 2017-01-11.
  • Published online: 2017-04-28.
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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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