The number of defective colorings of graphs on surfaces

A ( k , 1)‐coloring of a graph is a vertex‐coloring with k colors such that each vertex is permitted at most 1 neighbor of the same color. We show that every planar graph has at least c ρ n distinct (4, 1)‐colorings, where c is constant and ρ≈1.466 satisfies ρ 3 = ρ 2 + 1. On the other hand for any ε>0, we give examples of planar graphs with fewer than c (ϕ + ε) n distinct (4, 1)‐colorings, where c is constant and . Let γ( S ) denote the chromatic number of a surface S . For every surface S except the sphere, we show that there exists a constant c ′ = c ′( S )>0 such that every graph embeddable in S has at least c ′2 n distinct (γ( S ), 1)‐colorings. © 2010 Wiley Periodicals, Inc. J Graph Theory 28:129‐136, 2011