# Triangle-free graphs of tree-width t are ceil((t + 3)/2)-colorable

## TL;DR

This paper proves that triangle-free graphs with a given tree-width t can be colored with at most ceil((t + 3)/2) colors, establishing a tight bound and linking it to online coloring of path-width t graphs.

## Contribution

The paper introduces a tight upper bound on the chromatic number of triangle-free graphs based on their tree-width, and connects this to online coloring of graphs with bounded path-width.

## Key findings

- Bound of ceil((t + 3)/2) colors for triangle-free graphs of tree-width t.
- The bound is proven to be tight.
- Connection established between coloring tree-width t graphs and online coloring of path-width t graphs.

## Abstract

We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.