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The above inequality can be interpreted as saying that $\\mu(G) \\geq \\mu(O_n)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F. M. Dong, who in fact showed that, $$\\frac{P_G(q)}{P_G(q-1)} \\geq \\frac{q^n}{(q-1)^n}$$ for all $q \\geq n$. There are examples showing that this"},"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":"1502.06032","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-20T23:05:24Z","cross_cats_sorted":[],"title_canon_sha256":"03436538de961779dced5ab140f1ba8613111a31481d95e186d83ae2042f49e8","abstract_canon_sha256":"8716227c22b64bfbde355ff9dd02acd40ab3b35223b3b70fbd62068ff6cf46cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:38.231492Z","signature_b64":"+Wuwf/DGqF6hC99TTUtuahmqqAIUFMAi/HxoNm5k8VRHJ2WvvCHJLc10/CQeKw3zU0/33I59Yt4/aDxGqspPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d6bb36a929e9eaf0766cbb378ebb8c82cb707c526d2592e486ac138ab55b8b9","last_reissued_at":"2026-05-18T02:26:38.231089Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:38.231089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the shameful conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sukhada Fadnavis","submitted_at":"2015-02-20T23:05:24Z","abstract_excerpt":"Let $P_G(q)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The `shameful conjecture' due to Bartels and Welsh states that, $$\\frac{P_G(n)}{P_G(n-1)} \\geq \\frac{n^n}{(n-1)^n}.$$ Let $\\mu(G)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\\mu(G) \\geq \\mu(O_n)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F. M. Dong, who in fact showed that, $$\\frac{P_G(q)}{P_G(q-1)} \\geq \\frac{q^n}{(q-1)^n}$$ for all $q \\geq n$. 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