Abstract
Given an integer \(k>4\) and a graph H, we prove that, assuming P\(\ne \) NP, the List-k -Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if either every component of H is a path on at most three vertices, or removing the isolated vertices of H leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all \(k\ge 1\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: List coloring in the absence of a linear forest. Algorithmica 71(1), 21–35 (2015)
Edwards, K.: The complexity of colouring problems on dense graphs. Theoret. Comput. Sci. 43(2–3), 337–343 (1986)
Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of coloring graphs with forbidden subgraphs. J. Graph Theory 84(4), 331–363 (2017)
Hajebi, S., Li, Y., Spirkl, S.: Complexity dichotomy for list-5-coloring with a forbidden induced subgraph. SIAM J. Discrete Math. 36(3), 2004–2027 (2022)
Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10(4), 718–720 (1981)
Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)
Kamiński, M., Lozin, V.: Coloring edges and vertices of graphs without short or long cycles. Contrib. Discrete Math. 2(1), 61–66 (2007)
Karp, R.: Reducibility among combinatorial problems (1972). In: Ideas that Created the Future—Classic Papers of Computer Science, pp. 349–356. MIT Press, Cambridge (2021)
Leven, D., Galil, Z.: NP completeness of finding the chromatic index of regular graphs. J. Algorithms 4(1), 35–44 (1983)
Acknowledgements
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912]. This project was funded in part by the Government of Ontario. This research was conducted while Spirkl was an Alfred P. Sloan Fellow. Maria Chudnovsky is supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chudnovsky, M., Hajebi, S. & Spirkl, S. List-k-Coloring H-Free Graphs for All \(k>4\). Combinatorica (2024). https://doi.org/10.1007/s00493-024-00106-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00493-024-00106-2