Fall coloring of graphs I

A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number χf (G) of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, χ(G) ≤ χf (G). In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and χf (G). We also obtain the smallest non-fall colorable graphs with a given minimum degree δ and determine their number. These answer two of the questions raised by Dunbar et al.