Papers › New constructions for the n-queens problem
New constructions for the n-queens problem
Martin Bača, Susana-Clara López, Francesc-Antoni Muntaner-Batle, Andrea Semaničová-Feňovčíková
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Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function l:V(G)⟶{1,2,…,|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)≠l(u′)+l(v′) and (ii) l(v)-l(u)≠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 ⊗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 ⊗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 ⊗ₕ-product, which in some sense complements a previous result due to P\'olya.
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