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The Non-Biplanar Character of the Complete 9-Graph

  • W. T. Tutte (a1)

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Let us define a planar partition of a graph G as a pair {H1, H2} of subgraphs of G with the following properties

  1. (i)Each of H1 and H2 includes all the vertices of G.
  2. (ii)Each edge of G belongs to just one of H1 and H2.
  3. (iii)H1 and H2 are planar graphs.

It is not required that H1 and H2 are connected. Moreover either of these graphs may have isolated vertices, incident with none of its edges.

We describe a graph having a planar partition as biplanar.

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References

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1. Dirac, G. A. and Schuster, S., A theorem of Kuratowski, Proc. of the Nederl. Ak. W., 57(1954), 343-348.
2. Battle, J., Harary, F. and Kodama, Y., Every planar graph with nine points has a non-planar complement, Bull. Amer. Math. Soc, 68(1962), 569-571.
3. Kuratowski, C., Sur le problème des courbes gauches en Topologie, Fundam. Math., 15(1930), 271-283.
4. Tutte, W. T., A census of planar triangulations, Can. J. Math. 14 (1962), 21-38.
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The Non-Biplanar Character of the Complete 9-Graph

  • W. T. Tutte (a1)

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