{"paper":{"title":"Extremal problems of Erd\\H{o}s, Faudree, Schelp and Simonovits on paths and cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Binlong Li, Bo Ning, Jie Ma","submitted_at":"2021-02-08T17:20:24Z","abstract_excerpt":"For positive integers $n>d\\geq k$, let $\\phi(n,d,k)$ denote the least integer $\\phi$ such that every $n$-vertex graph with at least $\\phi$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erd\\H{o}s, Faudree, Schelp and Simonovits proposed the study of the function $\\phi(n,d,k)$, and conjectured that for any positive integers $n>d\\geq k$, it holds that $\\phi(n,d,k)\\leq \\lfloor\\frac{k-1}{2}\\rfloor\\lfloor\\frac{n}{d+1}\\rfloor+\\epsilon$, where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise. In this paper we determine the values of the function $\\phi(n,d,k)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.04367","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2102.04367/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}