# Equitable tree-$O(d)$-coloring of $d$-degenerate graphs

## TL;DR

This paper improves the bounds for equitable tree-coloring of $d$-degenerate graphs from exponential to linear in $d$, under certain size conditions, enhancing understanding of graph colorings with forest constraints.

## Contribution

The authors establish a linear bound on the number of colors needed for equitable tree-coloring of $d$-degenerate graphs, refining previous exponential bounds.

## Key findings

- Linear bounds for equitable tree-$k$-coloring established
- Coloring applies to graphs with size proportional to maximum degree
- Improves theoretical understanding of graph colorings with forest constraints

## Abstract

An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors so that every color class incudes a forest and the sizes of any two color classes differ by at most one.This kind of coloring was first introduced in 2013 and can be used to formulate the structure decomposition problem on the communication network with some security considerations. In 2015, Esperet, Lemoine and Maffray showed that every $d$-degenerate graph admits an equitable tree-$k$-coloring for every $k\geq 3^{d-1}$. Motivated by this result, we attempt to lower their exponential bound to a linear bound. Precisely, we prove that every $d$-degenerate graph $G$ admits an equitable tree-$k$-coloring for every $k\geq \alpha d$ provided that $|G|\geq \beta \Delta(G)$, where $(\alpha,\beta)\in \{(8,56), (9,26), (10,18), (11,15), (12,13), (13,12), (14,11), (15,10), (17,9), (20,8), (27,7), (52,6)\}$.