On the monatic number of a graph (original) (raw)

In a graph G AE (V,E), a set M µ V (G) is said to be a monopoly set of G if every vertex v 2 V iM has, at least, d(v) 2 neighbors in M. The monopoly size mo(G) is the minimum cardinality of a monopoly set. An M-partition of a graph G is the partition of the vertex set V (G) of G into k disjoint monopoly sets. The monatic number of G, denoted by ¹(G), is the maximum number of sets in M-partition of G. In this paper, we establish Nordhaus-Gaddum inequalities for monatic number of a graph. It is shown that, for any connected graph G with at least two edges, ¹(L(G)) AE 3 if and only if G AEC3k , where L(G) is the line graph of G and C3k is a cycle with 3k vertices, for k ¸ 1. The monatic numbers of the join G1 AG2 and corona G1 ±G2 of any two graph G1 and G2 are found. MSC: 05C70 ² 05C69