List-Recoloring of Sparse Graphs
Daniel W. Cranston

TL;DR
This paper proves that for certain classes of sparse graphs and list assignments, it is possible to transform one proper coloring into another with a bounded number of recolorings per vertex, confirming existing conjectures.
Contribution
It establishes bounds on recoloring sequences for triangle-free planar graphs and graphs with bounded maximum average degree, confirming conjectures by Dvořák and Feghali.
Findings
Recoloring sequences exist with bounded steps for triangle-free planar graphs.
Recoloring sequences exist with bounded steps for graphs with mad < 17/5 and 6-assignments.
Recoloring sequences exist with bounded steps for graphs with mad < 22/9 and 4-assignments.
Abstract
Fix a graph , a list-assignment for , and -colorings and . An -recoloring sequence, starting from , recolors a single vertex at each step, so that each resulting intermediate coloring is a proper -coloring. An -recoloring sequence transforms to if its initial coloring is and its final coloring is . We prove there exists an -recoloring sequence that transforms to and recolors each vertex at most a constant number of times if (i) is triangle-free and planar and is a 7-assignment, or (ii) and is a 6-assignment or (iii) and is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvo\v{r}\'{a}k and Feghali.
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Taxonomy
TopicsNuclear Receptors and Signaling
