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This paper proves the following results: (1) If $G$ is a connected $d$-degenerate graph, and $k>d$ is a prime number, and $G$ is either non-bipartite or has two non-adjacent ver"},"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":"1510.00809","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-03T12:15:15Z","cross_cats_sorted":[],"title_canon_sha256":"2e2926834cd26a618a63f8e11559ab208d2145af0c1d2fc42e585942252dd71e","abstract_canon_sha256":"d3ee7946ca2d44945efb9a616e03877f6b737f6226daa774b25dcacc5d4d0a62"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:05.613599Z","signature_b64":"T/J18UfBgmQwA5NAsYz7x2rRi2dgci4XiRORIgL/6Dm0tCQp+p/IWFFU9WHJNfzCNk+7iAEMnp69F7LnwqWICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8022fcf966a103cd259ff43fd28c0db145132f74d8d025eed1b2391382f57ab","last_reissued_at":"2026-05-18T01:31:05.612862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:05.612862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Total weight choosability of d-degenerate graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tsai-Lien Wong, Xuding Zhu","submitted_at":"2015-10-03T12:15:15Z","abstract_excerpt":"A graph $G$ is $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a total weighting\n  $\\phi: V(G) \\cup E(G) \\to R$ such that $\\phi(z) \\in L(z)$ for $z \\in V \\cup E$, and $\\sum_{e \\in E(u)}\\phi(e)+\\phi(u) \\ne \\sum_{e \\in E(v)}\\phi(e)+\\phi(v)$ for every edge $uv$. 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