Graph recoloring is the transformation of one proper vertex coloring of a graph into another by changing the color
of one graph vertex at a time while keeping every
intermediate coloring proper. The 
-reconfiguration graph 
has the proper vertex
colorings of 
using 
available colors as vertices, with two adjacent iff they differ at exactly one graph
vertex. A graph 
is recolorable if 
is connected for every
, where 
is the chromatic number.
Huang et al. (2026) proved that every clique blowup of the cycle graph 
is recolorable iff 
is 3, 4, 5, or 7. Here a clique blowup replaces each graph
vertex by a nonempty clique and joins two replacement
cliques completely precisely when their original vertices
were adjacent. This differs from the independent
vertex set replacement used in a graph blow-up.
