Partitioning edges of a planar graph into linear forests and a matching

Abstract

We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $\Delta \ge 9$ can be partitioned into $\frac{\Delta -1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $\Delta$ [Vizing, 1965] and linear arboricity at most $\lceil (\Delta +1)/2\rceil$ [Wu, 1999].