{"paper":{"title":"Improved Upper Bounds on $a'(G\\Box H)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Devanshi Vyas, Gaurav Patel, Om Thakkar, Punit Mehta, Rahul Muthu","submitted_at":"2015-07-07T14:08:08Z","abstract_excerpt":"The acyclic edge colouring problem is extensively studied in graph theory. The corner-stone of this field is a conjecture of Alon et. al.\\cite{alonacyclic} that $a'(G)\\le \\Delta(G)+2$. In that and subsequent work, $a'(G)$ is typically bounded in terms of $\\Delta(G)$. Motivated by this we introduce a term $gap(G)$ defined as $gap(G)=a'(G)-\\Delta(G)$. Alon's conjecture can be rephrased as $gap(G)\\le2$ for all graphs $G$. In \\cite{manusccartprod} it was shown that $a'(G\\Box H)\\le a'(G)+a'(H)$, under some assumptions. Based on Alon's conjecture, we conjecture that $a'(G\\Box H)\\le a'(G)+\\Delta(H)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01818","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"}