  A path decomposition of a graph $G$ is a collection of edge-disjoint paths of
$G$ that covers the edge set of $G$. Gallai (1968) conjectured that every
connected graph on $n$ vertices admits a path decomposition of cardinality at
most $\lfloor (n+1)/2\rfloor$. Gallai's Conjecture has been verified for many
classes of graphs. In particular, Lov\'asz (1968) verified this conjecture for
graphs with at most one vertex with even degree, and Pyber (1996) verified it
for graphs in which every cycle contains a vertex with odd degree. Recently,
Bonamy and Perrett (2016) verified Gallai's Conjecture for graphs with maximum
degree at most $5$, and Botler et al. (2017) verified it for graphs with
treewidth at most $3$. In this paper, we verify Gallai's Conjecture for
triangle-free planar graphs.
