On Fall Colorings of Graphs

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex of G sees all k colors on its closed neighborhood. We denote Fall(G) the set of all positive integers k for which G has a fall k-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of [3] about fall colorings of categorical product of complete graphs. Then, we study fall colorings of union of graphs. Then, we prove that fall k-colorings of a graph can be reduced into proper k-colorings of graphs in a specified set. Then, we characterize fall colorings of Mycielskian of graphs. Finally, we prove that for each bipartite graphG, Fall(Gc) ⊆ { χ(Gc) } and it is polynomial time to decision whether or not Fall(Gc) = { χ(Gc) } .