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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.
Our main result is that every graph $G$ on $nge 10^4r^3$ vertices with minimum degree $delta(G) ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, Kuhn, Lo and Osthus leads to the bes
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
We prove that for any integer $kgeq 2$ and $varepsilon>0$, there is an integer $ell_0geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+varepsilon)n$ has a fractional decomposition into tight cycles of le
Let $G$ be a graph whose edges are coloured with $k$ colours, and $mathcal H=(H_1,dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edg
It is proved that if a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor, then its line graph is Hamilton decomposable. This result partially extends Kotzigs result that a $3