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On List k-Coloring Convex Bipartite Graphs

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Graphs and Combinatorial Optimization: from Theory to Applications

Part of the book series: AIRO Springer Series ((AIROSS,volume 5))

Abstract

List k-Coloring (Lik -Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1, 2, …, k}. The problem is known to be NP-hard even for k = 3 within the class of 3-regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of Li 3-Col in the class of chordal bipartite graphs. In this paper, we give a partial answer to this question by showing that Lik -Col is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright et al. (SIAM J Discrete Math 28(4):1675–1685, 2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the Lik -Col in the class of convex bipartite graphs. Finally, we show how our algorithm can be modified to solve the more general LiH -Col problem on convex bipartite graphs.

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Acknowledgements

J. Díaz and M. Serna are partially supported by funds from MINECO and EU FEDER under grant TIN 2017-86727-C2-1-R and AGAUR project ALBCOM 2017-SGR-786. Ö. Y. Diner is partially supported by the Scientific and Technological Research Council Tübitak under project BIDEB 2219-1059B191802095 and by Kadir Has University under project 2018-BAP-08. O. Serra is supported by the Spanish Ministry of Science under project MTM2017-82166-P.

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Authors and Affiliations

  1. ALBCOM Research Group, CS Department, Universitat Politècnica de Catalunya, Barcelona, Spain

    Josep Díaz & Maria Serna

  2. Mathematics Department, Universitat Politècnica de Catalunya, Barcelona, Spain

    Öznur Yaşar Diner

  3. Computer Engineering Department, Kadir Has University, Istanbul, Turkey

    Öznur Yaşar Diner

  4. Mathematics Department, Universitat Politècnica de Catalunya, Barcelona, Spain

    Oriol Serra

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  1. Josep Díaz
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  2. Öznur Yaşar Diner
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  3. Maria Serna
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  4. Oriol Serra
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Correspondence to Josep Díaz .

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  1. Consiglio Nazionale delle Ricerce, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, Roma, Italy

    Claudio Gentile

  2. Consiglio Nazionale delle Ricerce, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, Roma, Italy

    Giuseppe Stecca

  3. Consiglio Nazionale delle Ricerce, Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, Roma, Italy

    Paolo Ventura

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Díaz, J., Diner, Ö.Y., Serna, M., Serra, O. (2021). On List k-Coloring Convex Bipartite Graphs. In: Gentile, C., Stecca, G., Ventura, P. (eds) Graphs and Combinatorial Optimization: from Theory to Applications. AIRO Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-63072-0_2

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