10-list Recoloring of Planar Graphs
Daniel W. Cranston

TL;DR
This paper proves that for any two list colorings of a planar graph with lists of size 10, there exists a bounded-length recoloring sequence transforming one into the other, confirming a conjecture and introducing a new reducibility technique.
Contribution
The paper establishes a universal bound on recoloring steps between any two 10-list colorings of planar graphs, confirming a key conjecture and presenting a novel reducibility method.
Findings
Existence of a constant C bounding recoloring steps
Recoloring sequences have length at most C|V(G)|
New technique for showing reducibility of configurations
Abstract
Fix a planar graph and a list-assignment with for all . Let and be -colorings of . A recoloring sequence from to is a sequence of -colorings, beginning with and ending with , such that each successive pair in the sequence differs in the color on a single vertex of . We show that there exists a constant such that for all choices of and there exists a recoloring sequence from to that recolors each vertex at most times. In particular, has length at most . This confirms a conjecture of Dvo\v{r}\'{a}k and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
