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It is well-known that the $k$-recolouring graph is connected for $k\\geq \\Delta+2$. Feghali, Johnson and Paulusma [Journal of Graph Theory, 83(4):340--358] showed that the $(\\Delta+1)$-recolouring graph is composed by a unique connected component of size at least $2$ and (possibly many) isolated vertices.\n  In this paper, we study the proportion of isolated verti"},"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":"1811.12650","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-30T07:34:36Z","cross_cats_sorted":[],"title_canon_sha256":"6f2c3183e70e3c5dc139a2c0c30dd5cdfecfd781d1177181998460a855b86764","abstract_canon_sha256":"77bdffd42433ac9e7451937148a3192ac27d39c41d1ede9dfb3f3de43f72e965"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:28.939151Z","signature_b64":"Z+M0tIBRPiL+cvdD4Sbfpg+UGNOQUKEAPVFNL1watuquqaJF/iIMmfDZ4kewGg1LjT33lxIcr3UsRMdEIHZUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7add614a46243be31c8d293c267bc6382acb7bb3acb431ee95e91ff7dd807bce","last_reissued_at":"2026-05-17T23:59:28.938511Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:28.938511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Frozen $(\\Delta+1)$-colourings of bounded degree graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guillem Perarnau, Marthe Bonamy, Nicolas Bousquet","submitted_at":"2018-11-30T07:34:36Z","abstract_excerpt":"Let $G$ be a graph of maximum degree $\\Delta$ and $k$ be an integer. 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