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  • Planar graph and map 3-colorability reduce to one another


    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.

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  • 原文地址:https://www.cnblogs.com/xhj928675426/p/14007801.html
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