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

## TL;DR

This paper proves that planar graphs with maximum degree up to 9 can have their edges partitioned into four linear forests and a matching, extending to higher degrees and strengthening existing edge-coloring and arboricity results.

## Contribution

It introduces a new partitioning method for planar graphs with maximum degree up to 9, improving understanding of their edge decompositions.

## Key findings

- Planar graphs with max degree ≤ 9 can be partitioned into 4 linear forests and a matching.
- For odd maximum degree ≥ 9, edges can be partitioned into (Δ-1)/2 linear forests and a matching.
- Strengthens known results on chromatic index and linear arboricity of planar graphs.

## 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 $\tfrac{\Delta-1}{2}$ 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].