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Linear Transformations Between Colorings in Chordal Graphs

Authors Nicolas Bousquet , Valentin Bartier

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  • Mathematics of computing → Graph theory
Keywords
  • graph recoloring
  • chordal graphs

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Abstract

Let k and d be such that k >= d+2. Consider two k-colorings of a d-degenerate graph G. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. 
If k=d+2, we know that there exists graphs for which a quadratic number of recolorings is needed. And when k=2d+2, there always exists a linear transformation. In this paper, we prove that, as long as k >= d+4, there exists a transformation of length at most f(Delta) * n between any pair of k-colorings of chordal graphs (where Delta denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two k-colorings c_1,c_2 computes a linear transformation between c_1 and c_2.

Cite As Get BibTex

Nicolas Bousquet and Valentin Bartier. Linear Transformations Between Colorings in Chordal Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ESA.2019.24

Author Details

Nicolas Bousquet
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, France
Valentin Bartier
  • Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, France

Funding

This work was supported by ANR project GrR (ANR-18-CE40-0032).

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