The strong equitable vertex 2-arboricity of complete bipartite and tripartite graphs

TL;DR

This paper determines the exact strong equitable vertex 2-arboricity for complete bipartite and tripartite graphs, providing precise values for these classes of graphs.

Contribution

It establishes the exact values of the strong equitable vertex 2-arboricity for complete bipartite and tripartite graphs, advancing understanding of equitable tree-colorings.

Findings

Exact values of $va^ ext{equiv}_2(K_{m,n})$ are determined.

Exact values of $va^ ext{equiv}_2(K_{l,m,n})$ are determined.

Results contribute to graph coloring theory and equitable partitioning.

Abstract

A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a $(q,r)$-tree-coloring such that the sizes of any two color classes differ by at most one. Let the \emph{strong equitable vertex $r$-arboricity} be the minimum $p$ such that $G$ has an equitable $(q, r)$-tree-coloring for every $q\geq p.$ In this paper, we find the exact value for each $va^\equiv_2(K_{m,n})$ and $va^\equiv_2(K_{l,m,n}).$