A matroid coloring assigns colors to the elements of the finite ground set of a loopless matroid so that every color class is independent. Its chromatic number is the least possible number of colors. For two loopless matroids on the same finite ground set, a common coloring requires each color class to be independent in both matroids.
In list coloring, each element 
must receive a color from its prescribed list 
. The common list chromatic number 
is the least 
for which every assignment of lists of size 
permits a common coloring (Garami 2026).
Garami (2026) constructed a graphic matroid 
and a partition
matroid 
with
 |
Thus equality of the chromatic and list chromatic numbers for one matroid does not extend to pairs of matroids. Garami (2026) also proved that two strongly base-orderable matroids, each admitting a coloring with two colors, have common list chromatic number at most 2.
