The clustered chromatic number of a nonempty class 
of finite simple
graphs with at least one graph vertex is the
least positive integer 
for which there is a positive
integer 
such that every graph in 
has a clustered coloring
with at most 
colors and with every monochromatic connected
component containing at most 
vertices (Grzesik et al.
2026, p. 2). The bound 
is uniform over the entire class. If no such 
exists, the clustered chromatic number is defined to be infinity.
For example, the class of all cycle graphs has clustered chromatic number 2. Two colors suffice with monochromatic connected components of size at most 2, while a single color cannot give a uniform bound for cycles of arbitrarily large size (Grzesik et al. 2026, p. 2).
