{"paper":{"title":"On hamiltonian cycles of 1-tough $(P_{2} \\cup kP_{1})$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Masahiro Sanka","submitted_at":"2026-05-19T08:06:58Z","abstract_excerpt":"Let $k$ be a positive integer. A graph is said to be $(P_2 \\cup kP_1)$-free if it does not contain $P_2 \\cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \\cup kP_1)$-free graph is hamiltonian or the Petersen graph. Note that this problem is affirmative for $k \\in \\{1,2,3\\}$ by the known results. In this paper, we show that for each integer $k \\geq 4$, if $G$ is a $1$-tough and $(k-1)$-connected $(P_2 \\cup kP_1)$-free graph with $|V(G)| \\ge k^2+k+1$ and $\\delta(G) \\ge k$, then $G$ is hamiltonian. This result implies that the abov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19508","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.19508/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"}