Minimal Steiner trees for rectangular arrays of lattice points
By distinction of several cases, the authors are able to construct a minimal Steiner tree for an arbitrary rectangular array of integer lattice points in the plane. For \(n\times n\)-arrays, this proves a conjecture by \textit{F. Chung}, \textit{M. Gardner} and \textit{R. Graham} [Math. Mag. 62, No. 2, 83-96 (1989; Zbl 0681.05018)] with the exception of the case \(n\equiv 0\bmod 6\), \(n>6\), where the authors were able to improve the conjecture. The proof rests on a theorem of another paper by the same authors [J. Comb. Theory, Ser. A 78, No. 1, 51-91 (1997; Zbl 0874.05018)], which characterizes the full components of a minimal Steiner tree for somewhat more general lattice sets. For non-square rectangular arrays, the proof is rather involved, but many drawings help to understand the constructions.
scientific article; zbMATH DE number 1051477