{"paper":{"title":"On the clique number of the square of a line graph and its relation to Ore-degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Luke Postle, Maxime Faron","submitted_at":"2017-08-07T18:50:27Z","abstract_excerpt":"In 1985, Erd\\H{o}s and Ne\\v{s}et\\v{r}il conjectured that the square of the line graph of a graph $G$, that is $L(G)^2$, can be colored with $\\frac{5}{4}\\Delta(G)^2$ colors. This conjecture implies the weaker conjecture that the clique number of such a graph, that is $\\omega(L(G)^2)$, is at most $\\frac{5}{4}\\Delta(G)^2$. In 2015, \\'Sleszy\\'nska-Nowak proved that $\\omega(L(G)^2)\\le \\frac{3}{2}\\Delta(G)^2$. In this paper, we prove that $\\omega(L(G)^2)\\le \\frac{4}{3}\\Delta(G)^2$. This theorem follows from our stronger result that $\\omega(L(G)^2)\\le \\frac{\\sigma(G)^2}{3}$ where $\\sigma(G) := \\max_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02264","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"}