Linear transformations between colorings in chordal graphs

07/03/2019

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by   Nicolas Bousquet, et al.

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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(Δ) · n between any pair of k-colorings of chordal graphs (where Δ 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.