2019/07/03 by Bousquet, Nicolas, Bartier, Valentin · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.01863
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 c1,c2 computes a linear transformation between c1 and c2.