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Our main result concerns this quantity when $G$ is regular and bipartite. We find numbers $0 \\leq a^-(k) \\leq a^+(k) \\leq 1$ with the property that for all such $G$, with"},"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":"1101.0839","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-01-04T22:16:07Z","cross_cats_sorted":[],"title_canon_sha256":"3733c5993dabdb08f530cc3ee98b19e18a859e9728616e65856fff26b96969e3","abstract_canon_sha256":"71458e3c1a813e7f6de7e6c846229933ee47939e97bb8eed33ba10a5b93e061b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:42.594448Z","signature_b64":"r4btEWTvtyXznnhKf8tDPTLJ21iRO1yyS+15kPh0RmiALibZFlHiswbtQrzg/w6fPlEr1PWrbkx1E3Swo+eNBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34213846bcf3c43b887cbd45dea4940d9746202d8c1e9812577f42fd693427ab","last_reissued_at":"2026-05-18T03:53:42.593644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:42.593644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"H-colouring bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Galvin, John Engbers","submitted_at":"2011-01-04T22:16:07Z","abstract_excerpt":"For graphs $G$ and $H$, an {\\em $H$-colouring} of $G$ (or {\\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as proper colourings and independent sets.\n  For a given $H$, $k \\in V(H)$ and $G$ we consider the proportion of vertices of $G$ that get mapped to $k$ in a uniformly chosen $H$-colouring of $G$. 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