{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/new-constructions-for-the-n-queens-problem","title":"New constructions for the $n$-queens problem","arxiv_id":"1703.09942","date":"2017-03-29","proceeding":null,"authors":["Martin Bača","Susana-Clara López","Francesc-Antoni Muntaner-Batle","Andrea Semaničová-Feňovčíková"],"abstract":"Let $D$ be a digraph, possibly with loops. A queen labeling of $D$ is a bijective function $l:V(G)\\longrightarrow \\{1,2,\\ldots,|V(G)|\\}$ such that, for every pair of arcs in $E(D)$, namely $(u,v)$ and $(u',v')$ we have (i) $l(u)+l(v)\\neq l(u')+l(v')$ and (ii) $l(v)-l(u)\\neq l(v')-l(u')$. Similarly, if the two conditions are satisfied modulo $n=|V(G)|$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of $1$-regular digraphs and the solutions of the (modular) $n$-queens problem. The $\\otimes_h$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $\\otimes_h$-product and some particular families of graphs. In this paper, we study some families of $1$-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) $n$-queens problem in terms of the $\\otimes_h$-product, which in some sense complements a previous result due to P\\'olya.","url_abs":"https://arxiv.org/abs/1703.09942v4","url_pdf":"https://arxiv.org/pdf/1703.09942v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"new-constructions-for-the-n-queens-problem","repo_url":"https://github.com/tesselwait/N-Queens","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}