Minimal Steiner Trees for 2k×2k Square Lattices

  title={Minimal Steiner Trees for 2k×2k Square Lattices},
  author={M. Brazil and T. Cole and J. Rubinstein and Doreen A. Thomas and J. Weng and N. Wormald},
  journal={J. Comb. Theory, Ser. A},
We prove a conjecture of Chung, Graham, and Gardner (Math. Mag.62(1989), 83?96), giving the form of the minimal Steiner trees for the set of points comprising the vertices of a 2k×2ksquare lattice. Each full component of these minimal trees is the minimal Steiner tree for the four vertices of a square. 
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  • H. Pollak
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 1978
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  • SIAM J . Appl . Math .
  • 1968
Melzak, On the problem of Steiner
  • Canad. Math. Bull
  • 1961