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

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