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A {\\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \\uplus P_2 \\uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \\uplus \\cdots \\uplus V_r$ (for some $r$) such that, for every $i \\in \\{1,\\cdots,r\\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \\cap P_1| = |V_i \\cap P_2|$. Then the {\\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ suc"},"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":"1212.2308","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-11T06:22:07Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"6b1c09b3a5a8392bae68b8b00dddd6592a1eb28263389a8aa81edecaaca100f9","abstract_canon_sha256":"891a192ba099d59dd3a841ed0a8cdb7d1b31d44a32dadbcc4358570bacb1f112"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:32.735354Z","signature_b64":"4N4DhnoXIspUbyv+z0C96Yno7aIeg4mfUssyimcUfW+6Tpm/lwt6bg+EerNA2X2sqNs9ZSLOAlMFQWP8yAYZAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24570022b43690111a6999f74c2e79eadb8023cbd0df159aeb915bbeb54ea9ce","last_reissued_at":"2026-05-18T02:58:32.734805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:32.734805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the balanced decomposition number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Tadashi Sakuma","submitted_at":"2012-12-11T06:22:07Z","abstract_excerpt":"A {\\em balanced coloring} of a graph $G$ means a triple $\\{P_1,P_2,X\\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \\uplus P_2 \\uplus X$ and $|P_1|=|P_2|$. A {\\em balanced decomposition} associated with the balanced coloring $V(G)=P_1 \\uplus P_2 \\uplus X$ of $G$ is defined as a partition of $V(G)=V_1 \\uplus \\cdots \\uplus V_r$ (for some $r$) such that, for every $i \\in \\{1,\\cdots,r\\}$, the subgraph $G[V_i]$ of $G$ is connected and $|V_i \\cap P_1| = |V_i \\cap P_2|$. Then the {\\em balanced decomposition number} of a graph $G$ is defined as the minimum integer $s$ suc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2308","kind":"arxiv","version":3},"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":"1212.2308","created_at":"2026-05-18T02:58:32.734888+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.2308v3","created_at":"2026-05-18T02:58:32.734888+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2308","created_at":"2026-05-18T02:58:32.734888+00:00"},{"alias_kind":"pith_short_12","alias_value":"ERLQAIVUG2IB","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"ERLQAIVUG2IBCGTJ","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"ERLQAIVU","created_at":"2026-05-18T12:27:04.183437+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/ERLQAIVUG2IBCGTJTH3UYLTZ5L","json":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L.json","graph_json":"https://pith.science/api/pith-number/ERLQAIVUG2IBCGTJTH3UYLTZ5L/graph.json","events_json":"https://pith.science/api/pith-number/ERLQAIVUG2IBCGTJTH3UYLTZ5L/events.json","paper":"https://pith.science/paper/ERLQAIVU"},"agent_actions":{"view_html":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L","download_json":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L.json","view_paper":"https://pith.science/paper/ERLQAIVU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.2308&json=true","fetch_graph":"https://pith.science/api/pith-number/ERLQAIVUG2IBCGTJTH3UYLTZ5L/graph.json","fetch_events":"https://pith.science/api/pith-number/ERLQAIVUG2IBCGTJTH3UYLTZ5L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L/action/storage_attestation","attest_author":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L/action/author_attestation","sign_citation":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L/action/citation_signature","submit_replication":"https://pith.science/pith/ERLQAIVUG2IBCGTJTH3UYLTZ5L/action/replication_record"}},"created_at":"2026-05-18T02:58:32.734888+00:00","updated_at":"2026-05-18T02:58:32.734888+00:00"}