Chromatic index, treewidth and maximum degree

TL;DR

This paper explores a conjecture relating chromatic index, treewidth, and maximum degree, proving a fractional version and improving existing bounds for certain graphs.

Contribution

It introduces a new conjecture linking treewidth and maximum degree to chromatic index, and proves a fractional case along with an improved bound for specific graphs.

Findings

Proves the fractional version of the conjecture.

Shows graphs with treewidth k and max degree ≥ k + √k satisfy χ'(G)=Δ(G).

Improves Vizing's bound for graphs with treewidth ≥ 4 and max degree 2k-1.

Abstract

We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $\Delta(G)\geq k + \sqrt{k}$ satisfies $\chi'(G)=\Delta(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\geq 4$ and maximum degree $2k-1$ satisfies $\chi'(G)=\Delta(G)$, improving an old result of Vizing.