A note on defective colorings of graphs in surfaces

A graph is (m, k)‐colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. In a recent paper Cowen, Cowen, and Woodall proved that, for each compact surface S, there exists an integer k = k(S) such that every graph in S can be (4, k)‐colored. They also conjectured that the 4 could be replaced by 3. In this note we prove their conjecture.