{"paper":{"title":"Increasing paths in edge-ordered graphs: the hypercube and random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Florian Pfender, Jessica De Silva, Michael Tait, Theodore Molla, Troy Retter","submitted_at":"2015-02-10T23:09:08Z","abstract_excerpt":"An edge-ordering of a graph $G=(V,E)$ is a bijection $\\phi:E\\to\\{1,2,...,|E|\\}$. Given an edge-ordering, a sequence of edges $P=e_1,e_2,...,e_k$ is an increasing path if it is a path in $G$ which satisfies $\\phi(e_i)<\\phi(e_j)$ for all $i<j$. For a graph $G$, let $f(G)$ be the largest integer $\\ell$ such that every edge-ordering of $G$ contains an increasing path of length $\\ell$. The parameter $f(G)$ was first studied for $G=K_n$ and has subsequently been studied for other families of graphs. This paper gives bounds on $f$ for the hypercube and the random graph $G(n,p)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03146","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"}