New Constructions for the n-Queens Problem

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) $l(u)+l(v)\ne l(u')+l(v')$ and (ii) $l(v)-l(u)\ne l(v')-l(u')$. 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 Pólya.