Great Circles in General Position II


The poles of three great circles define always a not-degenerated triangle;
all great circles of the arrangement define a spherical mosaik of 2+n*(n-1) facets.

Question:

 Is there a 3-coloring of the mosaic's vertices ?
I.e.: How many colors do we need if we assign each vertex a color so
that neighboured vertices have different colors ? Do we need just 3 colors ?

Conjecture:

 Conjecture: yes, there is always a 3-coloring of the arrangement

Remarks:

  • The mosaic's facets can be colored with two different colors
  • On each great circle there is an even amount of vertices ;
    thus we need only two colors for coloring the vertices on one great circle.
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    Literature: