{"paper":{"title":"Span capacities of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The d-capacity counts the maximum players who can simultaneously traverse every vertex of a graph while staying at least d apart, reaching the theoretical maximum for d=1 exactly when the graph is topfull.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aljo\\v{s}a \\v{S}uba\\v{s}i\\'c, Andrej Taranenko, Christopher Mouron, Mateja Gra\\v{s}i\\v{c}, Tanja Vojkovi\\'c","submitted_at":"2026-05-16T07:32:36Z","abstract_excerpt":"The $d$-capacity of a graph $G$ is introduced as the maximum number of players that can simultaneously traverse $G$ such that each player visits all vertices while maintaining a distance of at least $d$ under various movement rules. We determine their values for paths and cycles and provide bounds for bipartite graphs. Furthermore, we characterize topfull graphs, where the 1-capacities reach their theoretical maximum, establishing a connection to graph factorizations and connectivity."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize topfull graphs, where the 1-capacities reach their theoretical maximum, establishing a connection to graph factorizations and connectivity.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The movement rules under which players traverse the graph are assumed to be defined so that a finite maximum number of simultaneous traversals exists and can be attained for the graph classes studied (paths, cycles, bipartite graphs, and topfull graphs).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Defines d-capacity for graphs, computes exact values for paths and cycles, gives bounds for bipartite graphs, and characterizes topfull graphs via factorizations and connectivity.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The d-capacity counts the maximum players who can simultaneously traverse every vertex of a graph while staying at least d apart, reaching the theoretical maximum for d=1 exactly when the graph is topfull.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"97658556256f6d2e06fef44e764320596c32129087c61ffe5c306de4746c67bb"},"source":{"id":"2605.16852","kind":"arxiv","version":1},"verdict":{"id":"2960ca8c-8678-4ca9-a1d4-cbc527e5bca9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:46:30.713946Z","strongest_claim":"We characterize topfull graphs, where the 1-capacities reach their theoretical maximum, establishing a connection to graph factorizations and connectivity.","one_line_summary":"Defines d-capacity for graphs, computes exact values for paths and cycles, gives bounds for bipartite graphs, and characterizes topfull graphs via factorizations and connectivity.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The movement rules under which players traverse the graph are assumed to be defined so that a finite maximum number of simultaneous traversals exists and can be attained for the graph classes studied (paths, cycles, bipartite graphs, and topfull graphs).","pith_extraction_headline":"The d-capacity counts the maximum players who can simultaneously traverse every vertex of a graph while staying at least d apart, reaching the theoretical maximum for d=1 exactly when the graph is topfull."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16852/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T21:01:25.061703Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T21:01:19.240758Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.312755Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.386798Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c5b66166f9d6dd4ab3f05840783ba7a56e81e9ac734628f25a46039d23e96302"},"references":{"count":12,"sample":[{"doi":"","year":2023,"title":"Baniˇ c and A","work_id":"9e9149e9-b5b8-4c30-9c6e-a11f91f65ac1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"T. 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