Signed list edge coloring in graphs of bounded treewidth

Abstract

Vizing conjectured that the list edge chromatic number of any graph with maximum degree $\Delta$ is at most $\Delta + 1$. This conjecture has been confirmed for several important classes of graphs, in particular, Lang proved that it holds for all graphs of treewidth $3$. In this paper, we introduce the list edge coloring of signed graphs, a framework that generalizes both classical list edge coloring and the signed edge coloring introduced by Behr. We extend Lang's result by proving the signed analogue of Vizing's conjecture for all signed graphs of treewidth $3$, as well as for signed graphs of treewidth $4$ with maximum degree $\Delta \ge 10$.