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A path (resp. cycle) decomposition of a graph $G$ is a set of edge-disjointpaths (resp. cycles) of $G$ that covers the edge set of $G$. Gallai (1966)conjectured that every graph on $n$ vertices admits a path decomposition ofsize at most $\lfloor (n+1)/2\rfloor$, and Haj\'os (1968) conjectured thatevery Eulerian graph on $n$ vertices admits a cycle decomposition of size atmost $\lfloor (n-1)/2\rfloor$. Gallai's Conjecture was verified for manyclasses of graphs. In particular, Lov\'asz (1968) verified this conjecture forgraphs with at most one vertex of even degree, and Pyber (1996) verified it forgraphs in which every cycle contains a vertex of odd degree. Haj\'os'Conjecture, on the other hand, was verified only for graphs with maximum degree$4$ and for planar graphs. In this paper, we verify Gallai's and Haj\'os'Conjectures for graphs with treewidth at most $3$. Moreover, we show that theonly graphs with treewidth at most $3$ that do not admit a path decompositionof size at most $\lfloor n/2\rfloor$ are isomorphic to $K_3$ or $K_5-e$.Finally, we use the technique developed in this paper to present new proofs forGallai's and Haj\'os' Conjectures for graphs with maximum degree at most $4$,and for planar graphs with girth at least $6$.

On Gallai's and Haj\'os' Conjectures for graphs with treewidth at most 3