In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${rm ex}(T)$ paths, where ${rm ex}(T):= frac{1}{2}sum_{vin V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for all sufficiently
large tournaments. We also prove an asymptotically optimal result for tournaments of odd order.
We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $leftlceil frac{n}{2}rightrceil$ paths, while
a conjecture of Haj{o}s states that any Eulerian graph on $n$ vertices can be decomposed into at most $leftlfloor frac{n-1}{2}rightrfloor$ cycles. The ErdH{o}s-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $max left{frac{odd(G)}{2},frac{Delta(G)}{2}right}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $frac{Delta(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.
