Finding Domatic Partitions in Infinite Graphs

Abstract

We investigate the apparent difficulty of finding domatic partitions in graphs using tools from computability theory.  We consider nicely presented (i.e., computable) infinite graphs and show that even if the domatic number is known, there might not be any algorithm for producing a domatic partition of optimal size.  However, we prove that smaller domatic partitions can be constructed if we restrict to regular graphs.  Additionally, we establish similar results for total domatic partitions.