# Equitable vertex arboricity conjecture holds for graphs with low degeneracy

## TL;DR

This paper proves that graphs with low degeneracy can be equitably colored with forests under certain degree conditions, confirming a conjecture in this area.

## Contribution

It establishes the equitable vertex arboricity conjecture for graphs with low degeneracy by proving a new coloring property under specific degree constraints.

## Key findings

- Every d-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for k ≥ (Δ+1)/2.
- The conjecture holds for graphs where Δ ≥ 9.818d.
- Provides theoretical validation for equitable tree-coloring in low degeneracy graphs.

## Abstract

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$ provided that $\Delta\geq 9.818d$, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.