{"paper":{"title":"Word-Representability of Shift Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"All shift graphs and their multi-shift generalizations are word-representable.","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Ramesh Hariharasubramanian, Suchanda Roy","submitted_at":"2026-05-04T06:21:59Z","abstract_excerpt":"A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\\dots,x_k)$ with $1\\le x_1<x_2<\\cdots<x_k\\le n$, where two vertices $(x_1,\\dots,x_k)$ and $(y_1,\\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\\le i\\le k-1$ or $y_{i+1}=x_i$ for all $1\\le i\\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a single word over the vertex alphabet can be arranged so the alternation condition holds simultaneously for every pair of adjacent and non-adjacent k-tuples under the given shift rules.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Shift graphs G(n,k) and generalized shift graphs are word-representable.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"All shift graphs and their multi-shift generalizations are word-representable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b1787627b477539172acf0a135f8c018556d23cb7134ccf84b2204976d33530c"},"source":{"id":"2605.02268","kind":"arxiv","version":2},"verdict":{"id":"13d0a59f-8b64-4b83-b0b1-347030d4264d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T18:35:55.403098Z","strongest_claim":"we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable.","one_line_summary":"Shift graphs G(n,k) and generalized shift graphs are word-representable.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a single word over the vertex alphabet can be arranged so the alternation condition holds simultaneously for every pair of adjacent and non-adjacent k-tuples under the given shift rules.","pith_extraction_headline":"All shift graphs and their multi-shift generalizations are word-representable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.02268/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T16:35:15.496541Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T03:31:22.995004Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:31:07.893280Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"385669cf086d7f78b254890d32dae27d021ace883d10ceb27935d34ea8585161"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ab633ea5588f05ef5e899b8edfc39486b7cc9bdfaaa620ea97eab56c2ac318f8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}