New results on upper domatic number of graphs

Abstract

For a graph G = (V, E), a partition π = {V1, V2, . . . , Vk} of the vertex set V is an upper domatic partition if Vi dominates Vj or Vj dominates Vi or both for every Vi, Vj ∈ π, whenever i 6= j. The upper domatic number D(G) is the maximum order of an upper domatic partition of G. We study the properties of upper domatic number and propose an upper bound in terms of clique number. Further, we discuss the upper domatic number of certain graph classes including unicyclic graphs and power graphs of paths and cycles.

Keywords

• domination
• upper domatic partition
• upper domatic number
• transitivity