{"paper":{"title":"Toward \\.Zak's conjecture on graph packing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, Andrew McConvey, Derrek Yager, Ervin Gy\\H{o}ri","submitted_at":"2015-08-14T22:34:10Z","abstract_excerpt":"Two graphs $G_{1} = (V_{1}, E_{1})$ and $G_{2} = (V_{2}, E_{2})$, each of order $n$, pack if there exists a bijection $f$ from $V_{1}$ onto $V_{2}$ such that $uv \\in E_{1}$ implies $f(u)f(v) \\notin E_{2}$. In 2014, \\.{Z}ak proved that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$ and $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 96n^{3/4} - 65$, then $G_{1}$ and $G_{2}$ pack. In the same paper, he conjectured that if $\\Delta (G_{1}), \\Delta (G_{2}) \\leq n-2$, then $|E_{1}| + |E_{2}| + \\max \\{ \\Delta (G_{1}), \\Delta (G_{2}) \\} \\leq 3n - 7$ is sufficient for $G_{1}$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03672","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"}