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Minimal Steiner trees for rectangular arrays of lattice points. (English) Zbl 0883.05038

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 F. Chung, M. Gardner and 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.

MSC:

05C05 Trees
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References:

[1] Brazil, M.; Cole, T.; Rubinstein, J. H.; Thomas, D. A.; Weng, J. F.; Wormald, N. C., Minimal Steiner trees for \(2^k\)×\(2^k\) Square Lattices, J. Combin. Theory, Series A, 73, 91-110 (1996) · Zbl 0844.05036
[3] Chung, F. R.K.; Gardner, M.; Graham, R. L., Steiner trees on a checkerboard, Math. Magazine, 62, 83-96 (1989) · Zbl 0681.05018
[4] Chung, F. R.K.; Graham, R. L., Steiner trees for ladders, Ann. Disc. Math., 2, 173-200 (1978) · Zbl 0384.05030
[5] Hwang, F. K.; Richards, D. S.; Winter, P., The Steiner Tree Problem. The Steiner Tree Problem, Annals of Discrete Mathematics, 53 (1992), North-Holland: North-Holland Amsterdam · Zbl 0774.05001
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