Graphs with bounded tree-width and large odd-girth are almost bipartite

TL;DR

This paper proves that graphs with bounded tree-width and sufficiently large odd-girth are nearly bipartite, having a circular chromatic number close to 2, which advances understanding of graph coloring properties.

Contribution

It establishes a bound on the circular chromatic number for graphs with bounded tree-width and large odd-girth, showing they are almost bipartite.

Findings

Graphs with bounded tree-width and large odd-girth have circular chromatic number at most 2+ε.

Existence of a threshold odd-girth g for given tree-width k and ε.

Graphs become nearly bipartite as odd-girth increases.

Abstract

We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$.