A packing coloring of a graph assigns positive integer colors to its vertices so that two distinct vertices
with color 
have graph distance greater than 
. The least number of colors is the packing chromatic number
(Goddard et al. 2008). More generally, for a tuple 
of positive
integers, an 
-packing
coloring partitions the vertex set into possibly empty
classes 
such that distinct vertices in 
have graph distance greater
than 
(Goddard and Xu 2012). Taking all 
gives an ordinary proper vertex
coloring.
Romaniello (2026) proved that every claw-free graph of maximum vertex degree at most 3 admits
a 
-packing
coloring. Romaniello (2026) also constructed an infinite family of connected bridgeless
claw-free cubic graphs
without a 
-packing
coloring, beginning with a graph on 36 vertices.
