Equitable vertex arboricity of graphs

TL;DR

This paper introduces bounds on the strong equitable vertex arboricity for various classes of graphs, including complete bipartite, planar, and outerplanar graphs, advancing understanding of equitable forest colorings.

Contribution

It provides sharp upper bounds for the strong equitable vertex arboricity of complete bipartite graphs and planar graphs with certain girth conditions, and proposes conjectures for broader classes.

Findings

$va^{ ext{equiv}}_{1,1}(K_{n,n})=O(n)$

$va^{ ext{equiv}}_{k, ext{infty}}(K_{n,n})=O(n^{1/2})$

Bounded $va^{ ext{equiv}}_{ ext{infty}, ext{infty}}(G)$ for planar and outerplanar graphs

Abstract

An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$ and diameter at most $d$. The minimum $t$ such that $G$ has an equitable $(t',k,d)$-tree-coloring for every $t'\geq t$ is called the strong equitable $(k,d)$-vertex-arboricity and denoted by $va^{\equiv}_{k,d}(G)$. In this paper, we give sharp upper bounds for $va^{\equiv}_{1,1}(K_{n,n})$ and $va^{\equiv}_{k,\infty}(K_{n,n})$ by showing that $va^{\equiv}_{1,1}(K_{n,n})=O(n)$ and $va^{\equiv}_{k,\infty}(K_{n,n})=O(n^{\1/2})$ for every $k\geq 2$. It is also proved that $va^{\equiv}_{\infty,\infty}(G)\leq 3$ for every planar graph $G$ with girth at least 5 and $va^{\equiv}_{\infty,\infty}(G)\leq 2$ for every planar graph $G$ with girth at least 6 and for every outerplanar graph. We conjecture that $va^{\equiv}_{\infty,\infty}(G)=O(1)$ for every planar graph and $va^{\equiv}_{\infty,\infty}(G)\leq \lceil\frac{\Delta(G)+1}{2}\rceil$ for every graph $G$.