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Complexity of C_k-Coloring in Hereditary Classes of Graphs

Authors Maria Chudnovsky, Shenwei Huang, Paweł Rzążewski, Sophie Spirkl, Mingxian Zhong

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • homomorphism
  • hereditary class
  • computational complexity
  • forbidden induced subgraph

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Abstract

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs.
We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw.

Cite As Get BibTex

Maria Chudnovsky, Shenwei Huang, Paweł Rzążewski, Sophie Spirkl, and Mingxian Zhong. Complexity of C_k-Coloring in Hereditary Classes of Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.31

Author Details

Maria Chudnovsky
  • Princeton University, Princeton, NJ 08544, USA
Shenwei Huang
  • College of Computer Science, Nankai University, Tianjin 300350, China
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland
Sophie Spirkl
  • Rutgers University, Piscataway, NJ 08854, USA
Mingxian Zhong
  • Lehman College, CUNY, Bronx, NY 10468, USA

Funding

  • Chudnovsky, Maria: Supported by NSF grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF-16-1-0404.
  • Huang, Shenwei: This research is supported by the National Natural Science Foundation of China (11801284).
  • Spirkl, Sophie: This material is based upon work supported by the National Science Foundation under Award No. DMS1802201.

Acknowledgements

We are grateful to anonymous reviewers for their comments that helped improve the presentation of the paper.

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