Abstract
Suppose that we are given a positive integer k, and a k-(vertex-)colouring 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 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 . 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 , 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.
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.