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We prove that $\\gamma(G)\\leq \\left(1+\\frac{2(\\Delta-1)}{\\Delta 2^{\\Delta}}\\right)\\gamma_e(G)$ for general $\\Delta\\geq 1$, and $\\gamma(G)\\leq \\left(\\frac{7}{6}-\\frac{1}{204}\\right)\\gamma_e(G)$ for $\\Delta=3$. Furthermore, we verify our conjecture for cubic claw-free graphs."},"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":"1906.10420","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T09:47:23Z","cross_cats_sorted":[],"title_canon_sha256":"d7cbe0318e51c602fa950f517dcf664e49b5ec9935de890ce9f473be7ba6370f","abstract_canon_sha256":"c173b6da460e4e451e97e9c30df1a25e9e1332c931b009168e5281c1b7f75a66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:17.066936Z","signature_b64":"s3aFZk7dVLwGX6pwSGwwwpgRrQop6GMyGdnyhTZdJ+eayBLhvyiCPCyqhAjyhNkNkiigHjUONkwnrAXrgmdvAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e037a165950f5ded9bd95008f6c0da5001e026a9dd2de8001eb6b28c351d5e3","last_reissued_at":"2026-05-17T23:41:17.066368Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:17.066368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Domination versus edge domination","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Elena Mohr, Julien Baste, Maximilian F\\\"urst, Michael A. Henning","submitted_at":"2019-06-25T09:47:23Z","abstract_excerpt":"We propose the conjecture that the domination number $\\gamma(G)$ of a $\\Delta$-regular graph $G$ with $\\Delta\\geq 1$ is always at most its edge domination number $\\gamma_e(G)$, which coincides with the domination number of its line graph. We prove that $\\gamma(G)\\leq \\left(1+\\frac{2(\\Delta-1)}{\\Delta 2^{\\Delta}}\\right)\\gamma_e(G)$ for general $\\Delta\\geq 1$, and $\\gamma(G)\\leq \\left(\\frac{7}{6}-\\frac{1}{204}\\right)\\gamma_e(G)$ for $\\Delta=3$. 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