# Equitable Colorings of $K\_4$-minor-free Graphs

## TL;DR

This paper proves that all K4-minor-free graphs with maximum degree Δ can be equitably colored with at least (Δ+3)/2 colors, confirming a conjecture and using decomposition trees instead of discharging methods.

## Contribution

It establishes a tight bound for equitable coloring of K4-minor-free graphs, confirming a conjecture through a novel approach using decomposition trees.

## Key findings

- Bound is tight for equitable coloring with (Δ+3)/2 colors.
- Decomposition trees are effective for analyzing K4-minor-free graphs.
- Conjecture by Zhang and Whu is confirmed.

## Abstract

We demonstrate that for every positive integer $\Delta$, every K\_4-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.