{"paper":{"title":"Irregular subgraph in a regular graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hehui Wu, Quanyu Tang, Tianyue Cao","submitted_at":"2026-07-07T16:25:49Z","abstract_excerpt":"A conjecture of Alon and Wei states that, for any $d$-regular graph $G$ with $n$ vertices, there exists a spanning subgraph $H$ such that for all $0\\le i\\le d$, we have $m(H, i)$, the number of vertices in $H$ with degree $i$, is between $\\frac{n}{d+1}-2$ and $\\frac{n}{d+1}+2$. We prove the conjecture for all fixed $d$ when $n$ is sufficiently large. More precisely, if $q=(q_0,\\ldots,q_d)$ satisfies $$\n  \\sum_{i=0}^d q_i=n,\\qquad\n  \\sum_{i=0}^d i q_i\\equiv 0\\pmod 2,\\qquad\n  \\left|q_i-\\frac{n}{d+1}\\right|\\le 1\n  \\quad (0\\le i\\le d), $$ then there is a spanning subgraph $H\\subseteq G$ such that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06465","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/2607.06465/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"}