Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 496 — #504
496
Chapter 12
Simple Graphs
P OR Q
N
T
F
P
Q
[h]
Figure 12.29
A 3-color OR-gate
(b) Let E be an n-variable propositional formula, and suppose E defines a truth
function f W fT; F g n ! fT; F g. Explain a simple way to construct a graph that is
a 3-color-f -gate.
(c) Explain why an efficient procedure for determining if a graph was 3-colorable
would lead to an efficient procedure to solve the satisfiability problem, SAT.
Problem 12.30.
The 3-coloring problem for planar graphs turns out to be no easier than the 3-
coloring problem for arbitrary graphs. This claim follows very simply from the
existence of a “3-color cross-over gadget.” Such a gadget is a planar graph whose
outer face is a cycle with four designated vertices u; v; w; x occurring in clockwise
order such that
1. Any assignment of colors to vertices u and v can be completed into a 3-
coloring of the gadget.
2. In every 3-coloring of the gadget, the colors of u and w are the same, and the
colors of v and x are the also same.