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 of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, . In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and . We also obtain the smallest non-fall colorable...