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In this paper we characterize the graphs $G$ for which $\\cn(G)=\\la(G)+1$. The case $\\la(G)=3$ was already solved by Alboulker {\\em et al.\\,} \\cite{AlboukerV2016}. We show that a graph $G$ with $\\la(G)=k\\geq 4$ satisfies $\\cn(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\\'os join."},"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":"1603.09187","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-30T13:42:27Z","cross_cats_sorted":[],"title_canon_sha256":"eb71c0b84cdc39bc4b92009d9ef8e57eb8047cc4ad745ed512f8e9bd859d6631","abstract_canon_sha256":"b653b164c822a67143936932bd30eb3ecd7b6794904e2536ff9f5e1ad4a0eedf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:01.217291Z","signature_b64":"03AQZTxDUFogzqu9Gi77mJqrKpmTwOX2JPyZdj8lzRjOmG+r9ukupSLNhoMe1sHtqNaiiIo/RZ3eMePZxoSNCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d493994be10f68a289ee361c95f5573aa95c86cbe85813b816f4aa0c290efd56","last_reissued_at":"2026-05-18T01:18:01.216657Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:01.216657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Brooks type theorem for the maximum local edge connectivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bjarne Toft, Michael Stiebitz","submitted_at":"2016-03-30T13:42:27Z","abstract_excerpt":"For a graph $G$, let $\\cn(G)$ and $\\la(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \\cite{Dirac53} implies that every graph $G$ satisfies $\\cn(G)\\leq \\la(G)+1$. In this paper we characterize the graphs $G$ for which $\\cn(G)=\\la(G)+1$. The case $\\la(G)=3$ was already solved by Alboulker {\\em et al.\\,} \\cite{AlboukerV2016}. We show that a graph $G$ with $\\la(G)=k\\geq 4$ satisfies $\\cn(G)=k+1$ if and only if $G$ contains a block which can be obtained from copies of $K_{k+1}$ by repeated applications of the Haj\\'os join."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09187","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.09187","created_at":"2026-05-18T01:18:01.216748+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.09187v1","created_at":"2026-05-18T01:18:01.216748+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.09187","created_at":"2026-05-18T01:18:01.216748+00:00"},{"alias_kind":"pith_short_12","alias_value":"2SJZSS7BB5UK","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2SJZSS7BB5UKFCPO","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2SJZSS7B","created_at":"2026-05-18T12:29:55.572404+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK","json":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK.json","graph_json":"https://pith.science/api/pith-number/2SJZSS7BB5UKFCPOGYOJL5KXHK/graph.json","events_json":"https://pith.science/api/pith-number/2SJZSS7BB5UKFCPOGYOJL5KXHK/events.json","paper":"https://pith.science/paper/2SJZSS7B"},"agent_actions":{"view_html":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK","download_json":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK.json","view_paper":"https://pith.science/paper/2SJZSS7B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.09187&json=true","fetch_graph":"https://pith.science/api/pith-number/2SJZSS7BB5UKFCPOGYOJL5KXHK/graph.json","fetch_events":"https://pith.science/api/pith-number/2SJZSS7BB5UKFCPOGYOJL5KXHK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK/action/storage_attestation","attest_author":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK/action/author_attestation","sign_citation":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK/action/citation_signature","submit_replication":"https://pith.science/pith/2SJZSS7BB5UKFCPOGYOJL5KXHK/action/replication_record"}},"created_at":"2026-05-18T01:18:01.216748+00:00","updated_at":"2026-05-18T01:18:01.216748+00:00"}