Planar graph and map 3-colorability reduce to one another

Posted 2022-01-09 mb61caa1c74a413

目录

• 1.1 证明
• 2.1 Planar 3-colorability is NP-complete
  • 2.1.1 平面3着色特点
  • 2.1.2 构造方法

Theorem. PLANAR-3-COLOR ≣ P PLANAR-MAP-3-COLOR.
(一个平面三着色问题是可以跟一个平面图三着色问题相互规约的！)

1.1 证明

2.1 Planar 3-colorability is NP-complete

证明：1.首先证明它是一个NP问题很好证明只要我们在多项式时间内找到其中一个解就行了。
2.我们可以将3着色问题规约到平面3着色问题。
3.给你任意一个3着色实例G我们都能构造一个平面3着色的实例，如果这个平面是可以被3着色的，那么G就是可2着色的。

2.1.1 平面3着色特点

每一个平面如果可以被3着色那么相反的角具有相同的颜色，依次可以推出其他颜色。
Lemma. W is a planar graph such that:
・In any 3-coloring of W, opposite corners have the same color.
・Any assignment of colors to the corners in which opposite corners have
the same color extends to a 3-coloring of W.
Pf. The only 3-colorings (modulo permutations) of W are shown below. ▪

2.1.2 构造方法

Construction. Given instance G of 3-COLOR, draw G in plane, letting edges
cross. Form planar Gʹ by replacing each edge crossing with planar gadget W.
Lemma. G is 3-colorable iff Gʹ is 3-colorable.
・In any 3-coloring of W, a ≠ aʹ and b ≠ bʹ.
・If a ≠ aʹ and b ≠ bʹ then can extend to a 3-coloring of W.