Sketch a 5 regular planar graph, G with $\chi(G)$ = 3.

The graph of the snub cuboctahedron is an example with 24 vertices:

3-colored snub cube

(drawing adapted from here; see also Mathworld for other nice drawings).

I found this by taking four copies of this fragment

                 \   \
          \   \   \   )
           \   \   \ /
        .---A---B---C----
       /   / \ / \ /
      /   /   C---A-----
     /   /   / \ /
    /   /   /   B--------
       /   /   / \
              /   `-----

and arranging them as the vertices of a tetrahedron, and it turns out the 3-coloring fits together! In the above drawing, each copy of this basic fragment is tinted green, and the faces that correspond to the faces of the original tetrahedron are tinted red.

Tags:

Graph Theory

Planar Graphs

Coloring

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