On Gallai's conjecture for series-parallel graphs and planar 3-trees

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most $\lfloor 5n/8 \rfloor$ paths, which is an improvement over the best previously known bound of $\lfloor 2n/3 \rfloor$.