Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 497 — #505
12.10. References 497
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[h]
Figure 12.30
A 3-color cross-over gadget.
Figure 12.30 shows such a 3-color cross-over gadget. 10
So to find a 3-coloring for any simple graph, simply draw it in the plane with
edges crossing as needed, and then replace each occurrence of an edge crossing by
a copy of the gadget as shown in Figure 12.31. This yields a planar graph which
has a 3-coloring iff the original graph had one.
(a) Prove that the graph in Figure 12.30 satisfies condition (1) by exhibiting the
claimed 3-colorings.
10 This gadget and reduction of 3-colorability to planar 3-colorability are due to Larry Stockmeyer
[43].
[h]
Figure 12.31
Replacing an edge-crossing with a planar gadget.