Nordhaus–Gaddum-Type Results for the Strong Equitable Vertex k-Arboricity of Graphs

Abstract

For a graph [Formula: see text] and positive integers [Formula: see text], [Formula: see text], a [Formula: see text]-tree-vertex coloring of [Formula: see text] refers to a [Formula: see text]-vertex coloring of [Formula: see text] satisfying every component of each induced subgraph generated by every set of vertices with the same color forms a tree with maximum degree not larger than [Formula: see text], and it is called equitable if the difference between the cardinalities of every pair of sets of vertices with the same color is at most [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of [Formula: see text], denoted by [Formula: see text], is defined as the least positive integer [Formula: see text] satisfying [Formula: see text], which admits an equitable [Formula: see text]-tree-vertex coloring for each integer [Formula: see text] with [Formula: see text]. The strong equitable vertex [Formula: see text]-arboricity of a graph is very useful in graph theory applications such as load balance in parallel memory systems, constructing timetables and scheduling. In this paper, we present the tight upper and lower bounds on [Formula: see text] for an arbitrary graph [Formula: see text] with [Formula: see text] vertices and a given integer [Formula: see text] with [Formula: see text], and we characterize the extremal graphs [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], respectively. Based on the above extremal results, we further obtain the Nordhaus–Gaddum-type results for [Formula: see text] of graphs [Formula: see text] with [Formula: see text] vertices for a given integer [Formula: see text] with [Formula: see text].