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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.
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 the
In 2006, Barat and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies o
A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $mathcal{D}$ of $G$ such that every subgraph $H in mathcal{D}$ is locally irregular. A graph is s
We prove that for $k in mathbb{N}$ and $d leq 2k+2$, if a graph has maximum average degree at most $2k + frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having at most $d$ edges.
Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d le 4$, we are interested in the