Abstract
An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.
Highlights
All graphs considered in this paper are finite, undirected and simple
An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one
We focus on the strong equitable vertex 2-arboricity of Kn,n
Summary
All graphs considered in this paper are finite, undirected and simple. For a real number x, d x e is the least integer not less than x and b x c is the largest integer not larger than x. Noting that the upper bound given in Lemma 1 is not very tight for some special graphs, Wu et al [3] commented that determining the strong equitable 1-arboricity for every Kn,n seems not to be an easy task.
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