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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.06465","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T16:25:49Z","cross_cats_sorted":[],"title_canon_sha256":"08e6a7dc17d1569536e62fc3b34fe4b6c2f1fe28a0626d0a2c6987205396113c","abstract_canon_sha256":"9a8f166c611180499d47e2f50c215d21de8d15a30bcd239d5fe2b8f00ff06648"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:27.095192Z","signature_b64":"2YTQHK6dtEH2fPQSxPUDijxkctBx6+60b+UIoPuLv7/nt4CAZULuoQ7Vw/euB6NPvnm3fcceZPa6m7sDiF5eDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24d584ac0bfffc984841a91e140387a31eaf97fd35d0727199ec0ede6bea3131","last_reissued_at":"2026-07-08T01:19:27.094713Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:27.094713Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.06465","created_at":"2026-07-08T01:19:27.094778+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.06465v1","created_at":"2026-07-08T01:19:27.094778+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06465","created_at":"2026-07-08T01:19:27.094778+00:00"},{"alias_kind":"pith_short_12","alias_value":"ETKYJLAL776J","created_at":"2026-07-08T01:19:27.094778+00:00"},{"alias_kind":"pith_short_16","alias_value":"ETKYJLAL776JQSCB","created_at":"2026-07-08T01:19:27.094778+00:00"},{"alias_kind":"pith_short_8","alias_value":"ETKYJLAL","created_at":"2026-07-08T01:19:27.094778+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM","json":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM.json","graph_json":"https://pith.science/api/pith-number/ETKYJLAL776JQSCBVEPBIA4HUM/graph.json","events_json":"https://pith.science/api/pith-number/ETKYJLAL776JQSCBVEPBIA4HUM/events.json","paper":"https://pith.science/paper/ETKYJLAL"},"agent_actions":{"view_html":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM","download_json":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM.json","view_paper":"https://pith.science/paper/ETKYJLAL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.06465&json=true","fetch_graph":"https://pith.science/api/pith-number/ETKYJLAL776JQSCBVEPBIA4HUM/graph.json","fetch_events":"https://pith.science/api/pith-number/ETKYJLAL776JQSCBVEPBIA4HUM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM/action/storage_attestation","attest_author":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM/action/author_attestation","sign_citation":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM/action/citation_signature","submit_replication":"https://pith.science/pith/ETKYJLAL776JQSCBVEPBIA4HUM/action/replication_record"}},"created_at":"2026-07-08T01:19:27.094778+00:00","updated_at":"2026-07-08T01:19:27.094778+00:00"}