Decremental optimization of vertex-colouring under the reconfiguration framework

Abstract

Suppose that we are given a positive integer k, and a k-(vertex-)colouring $f_0$ of a given graph G. Then we are asked to find a colouring of G using the minimum number of colours among colourings that are reachable from $f_0$ by iteratively changing a colour assignment of exactly one vertex while maintaining the property of k-colourings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where $k\le 3$. These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any $k\ge 4$, this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colours.

Keywords:

• Combinatorial reconfiguration
• graph algorithms
• graph colouring
• linear-time algorithms

Acknowledgments

We are grateful to Tatsuhiko Hatanaka, Takehiro Ito and Haruka Mizuta for valuable discussions with them. We thank the anonymous referees for their constructive suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Akira Suzuki was partially supported by JSPS KAKENHI [grant numbers JP18H04091, JP20K11666 and JP20H05794], Japan. Yuma Tamura was partially supported by JSPS KAKENHI [grant number JP21K21278], Japan. Xiao Zhou was partially supported by JSPS KAKENHI [grant number JP19K11813], Japan.