On Fall-Colorable Graphs

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V (G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n (k−1) +r (k−r) ) /2, where r≡n (modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈5, 6.