{"paper":{"title":"Leaves for packings with block size four","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Peter J. Dukes, Tao Feng, Yanxun Chang","submitted_at":"2019-05-29T00:55:39Z","abstract_excerpt":"We consider maximum packings of edge-disjoint $4$-cliques in the complete graph $K_n$. When $n \\equiv 1$ or $4 \\pmod{12}$, these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible `leave' graphs induced by those edges. We give particular emphasis to the case $n \\equiv 0$ or $3 \\pmod{12}$, when the leave is $2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of $2$-regular leaves. For va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.12151","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"}