Improved Decomposition Bounds for Partition Polytopes and Odd-Covers

The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $k_1,\dots ,k_n$ can be encoded as the vertices of the partition polytope $\mathrm{P}\mathrm{P}(k_1,\dots ,k_n)$. We prove that, if $K=\max \{k_1,\dots ,k_n\}$, then the combinatorial diameter of $\mathrm{P}\mathrm{P}(k_1,\dots ,k_n)$ is at most $\lceil 3K\mathrm{/}2\rceil$. This improves the previously known upper bound of $2K$. A cycle (or path) odd-cover of a graph $G$ is a set of cycles (or paths) with symmetric difference $G$. We prove that every Eulerian graph $G$ with maximum degree $\mathrm{\Delta }$ admits a cycle odd-cover and a path odd-cover, each of size at most $\lceil 3\mathrm{\Delta }\mathrm{/}4\rceil$. This improves the previously known upper bound of $\mathrm{\Delta }$. The two proofs share many similarities and are both based on the proof of Akiyama, Exoo, and Harary that every graph with maximum degree 4 has linear arboricity at most 3.