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He proved that for any graph $G$,\n  \\begin{align} \\label{abstract} \\chi(G) \\geq conn(\\N(G))+3.\n  \\end{align}\n  In this article we show that for a class of exponential graphs the bound given in (\\ref{abstract}) is sharp. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy.\n  We were also able to find a class of exponential graphs, which are homotopy test graphs.\n  Hedetniemi's conjecture states that the chromatic number of the ca"},"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":"1810.00648","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-01T12:18:17Z","cross_cats_sorted":[],"title_canon_sha256":"197a29b63f12c08d2b64b000db7c2600618ff48d5bff09202dcaebe43e78a4f3","abstract_canon_sha256":"b4d1235b8d716ec2c25a8245d6ea2c918d3cb05c3e2b1728a776f4c4b179759b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:14.632351Z","signature_b64":"sxRCfZdukhtzY22LBmEzKMyq+nyfQNq6Wbo9RLSXw5H7TzaLajv09IGoW4yMYfPQINOIVInKANhPLk1ogsz1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af7becc1fc691c88a049ca83160ef48d23eceaadf674f39e1bb4f3744b7a1887","last_reissued_at":"2026-05-18T00:03:14.631704Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:14.631704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Samir Shukla","submitted_at":"2018-10-01T12:18:17Z","abstract_excerpt":"The neighborhood complex $\\N(G)$ of a graph $G$ were introduced by L. 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