Chromatic index, treewidth and maximum degree

Abstract

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

14 pages, 3 figures, minor changes, accepted for publication in Electronic Journal of Combinatorics