A group of operations on a partially colored map
A group of operations on a partially colored map
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部分彩色地图上的一组操作
DOI：
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发表时间：
1935
期刊：
影响因子：
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通讯作者：
I. Kittell
中科院分区：
文献类型：
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作者：
I. Kittell
前面已经证明了 *，如果有一个最小的不可着色的地图，并且除了一个五边区域用四种颜色A、B、C和D着色之外，它是着色的，那么只有一种颜色（比如B）将在与未着色区域接触的五个区域中重复，并且可以按顺序将颜色DBABC分配给这个区域环。同样，当地图被这样着色时，将有两条相交的肯普链分别从A到C和A到Dy。我们可以说，任何地图，无论最小不可着色与否，只要以上述方式部分着色，就是着色的绝境。这种情况可以如图1所示。图1该图并不试图示出电路在何处或彼此交叉多少次。这仅仅意味着他们交叉。在这里，我们将未着色的五边区域放置在地图的其余部分之外。我们这样做是为了简洁，但必须注意的是，下面的内容并不取决于这种表示的选择，也不取决于字母的选择。还要记住的事实是，右和左只有任意的意义，在讨论一个地图，可以在任何连续的方式变形的表面上的一个领域。大多数作家会画这样一幅地图，里面是无色的区域。我们在这里称D区域为上图中A右边的区域，C区域为A左边的区域。我们做如下定义。区域A称为环的顶点。它是该环中位于具有相同颜色的两个区域之间的区域。我们称交流电路为左手电路，称AD电路为右手电路。我们把在A和C之间的环中包括B区的BD链称为左手链
It has been previously shown* that if there be a minimum uncolorable map, and it is colored with the exception of one five-sided region in the four colors A, B, C, and D, then only one of the colors (say B) will be repeated in the five regions touching the region not colored, and that this ring of regions can be assigned the colors DBABC in order. I t has likewise been shown that when the map is thus colored, there will be two intersecting Kempe chains running from A to C and A to Dy respectively. We shall say that any map, whether minimum uncolorable or not, which is partially colored in the above manner is colored impasse. The situation may be represented as in Fig. 1. FIG. 1 The diagram does not at tempt to show where or how many times the circuits cross one another. I t merely means that they do cross. Here we have placed the uncolored five-sided region outside the rest of the map. We do this for compactness, but it must be noted that what follows is not dependent upon this choice of representation, nor upon the choice of lettering. Also to be remembered is the fact that right and left have only arbitrary meanings in discussing a map which may be deformed in any continuous manner over the surface of a sphere. Most writers would draw such a map with the uncolored region inside. We here call the D region the one in the diagram above on the right of A, and the C region the one on the left of A. We make the following definitions. The region A is called the vertex of the ring. It is the region in this ring which lies between the two regions having the same color. We call the AC circuit the left-hand circuit and the AD circuit the right-hand circuit. We call the BD chain which includes the B region in the ring between A and C the left-hand