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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.
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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.
We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal construction. It is an unbalanced blow-up of an edge, where the sink vertex is replaced by a quasi-random tournament and the source vertex is iteratively replaced by a copy of the construction itself.
In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Gir~ao. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $lceil 2n/3 rceil-1$, which is best possible.
We consider a generalisation of Kellys conjecture which is due to Alspach, Mason, and Pullman from 1976. Kellys conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kuhn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.
We describe computer searches that prove the graph reconstruction conjecture for graphs with up to 13 vertices and some limited classes on larger sizes. We also investigate the reconstructability of tournaments up to 13 vertices and posets up to 13 points. In all cases, our proofs also apply to the set reconstruction problem that uses the isomorph-reduced deck.
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