{"paper":{"title":"Odd decompositions of eulerian graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Edita M\\'a\\v{c}ajov\\'a, Martin \\v{S}koviera","submitted_at":"2016-06-30T21:30:50Z","abstract_excerpt":"We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\\equiv |E(G)|\\pmod{2}$. We conjecture that a connected $2d$-regular graph of odd order with $d\\ge 1$ admits a decomposition into $d$ odd closed trails sharing a common vertex and verify the conjecture for $d\\le 3$. The case $d=3$ is crucial for determining the flow number of a signed eulerian graph which is treated in a separate paper (arXiv:1408.1703v2). The proof of our conjecture for $d=3$ is surprisingly diffic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00053","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}