{"paper":{"title":"A merging procedure for labelings of bipartite graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"A merging procedure produces (A,B)-uniformly ordered labelings for bipartite graphs built by iteratively adding even cycles and pendant paths.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lucia Marino, Paola Bonacini","submitted_at":"2026-05-13T14:43:31Z","abstract_excerpt":"Let $G$ a bipartite graph with vertex bipartition $\\{A,B\\}$ and let $m=|E(G)|$. An $(A,B)$-uniformly ordered labeling of $G$ is a labeling $f\\colon V\\rightarrow [0,2m]$ which, among other conditions, requires that there exists $\\lambda\\in \\mathbb N$ such that $f(a)\\le \\lambda$ and $f(b)>\\lambda$ for all $a\\in A$ and $b\\in B$. The existence of such a labeling for $G$ implies the existence of a cyclic $G$-decomposition of $K_{2mx+1}$ for all positive integers $x$. In this paper, as a starting point, through this type of labeling we prove the existence of a cyclic $G$-decomposition in the case th"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Through a merging procedure, we are able to get this type of labeling for a specific class of bipartite graphs, which are obtained by iteratively adding an even cycle and a pendant path.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the merging operation on two (A,B)-uniformly ordered labelings always produces a valid labeling on the combined graph without violating the separation condition or the range requirements.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A merging procedure produces (A,B)-uniformly ordered labelings for a class of bipartite graphs formed by iteratively adding even cycles and pendant paths, yielding cyclic G-decompositions of K_{2mx+1}.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A merging procedure produces (A,B)-uniformly ordered labelings for bipartite graphs built by iteratively adding even cycles and pendant paths.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5c0cc598618cbf7f8e65720682a772a241d989abbc81f9e9c71d084056ff4cd9"},"source":{"id":"2605.13610","kind":"arxiv","version":1},"verdict":{"id":"c4438964-d3d2-4990-95ce-00a0765e9720","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:05:54.024545Z","strongest_claim":"Through a merging procedure, we are able to get this type of labeling for a specific class of bipartite graphs, which are obtained by iteratively adding an even cycle and a pendant path.","one_line_summary":"A merging procedure produces (A,B)-uniformly ordered labelings for a class of bipartite graphs formed by iteratively adding even cycles and pendant paths, yielding cyclic G-decompositions of K_{2mx+1}.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the merging operation on two (A,B)-uniformly ordered labelings always produces a valid labeling on the combined graph without violating the separation condition or the range requirements.","pith_extraction_headline":"A merging procedure produces (A,B)-uniformly ordered labelings for bipartite graphs built by iteratively adding even cycles and pendant paths."},"references":{"count":5,"sample":[{"doi":"","year":2012,"title":"K. 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Combin., (2025), 805 pp","work_id":"2853229b-ee2e-42a7-9acf-8964c48a5cae","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1966,"title":"Rosa,On certain valuations of the vertices of a graph, in:Th´ eorie des graphes, journ´ ees internationales d’´ etudes, Rome 1966 (Dunod, Paris, 1967), 349–355","work_id":"6dc8502b-246e-4598-aed8-7e51a280ec7b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":5,"snapshot_sha256":"e0436930fa8613090f1c172ed9e0c8feba0da43cc1a648606beedec72f172cf8","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"}