数学代写|图论作业代写Graph Theory代考| The shading of one section of the B-R

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数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

Since each section of a Kempe chain is isolated from the other sections of the same color pair, the colors of any section of a Kempe chain may be reversed and still satisfy the four-color theorem. This is an important and useful concept.

The shading of one section of the B-R chains above illustrates how the colors of any section of any Kempe chain may be reversed. Note that we reversed the colors of one section of the B-R chains, but didn’t reverse the colors of the section in the center. Each section of the same chain may have its colors reversed independently of the other sections of that chain.

Why do PG’s have Kempe chains? It’s easy to understand why MPG’s have Kempe chains. (Since a PG is formed by removing edges from a MPG and since the coloring that works for the MPG also works for the PG, it follows that PG’s also have Kempe chains.)

• A MPG is triangulated. It consists of faces with three edges and three vertices.
• The three vertices of each face must be three different colors.
• Each edge is shared by two adjacent triangles, which form a quadrilateral.
• Each quadrilateral will have 3 or 4 different colors. It has 3 colors if the two vertices opposite to the shared edge happen to be the same color.
• For each quadrilateral, at least 1 vertex and at most 3 of the four vertices have colors for any color pair. For example, a quadrilateral with R, G, B, and $G$ has 1 vertex with $R-Y$ and 3 vertices with $B-G$, or you could think of it as 1 vertex with $B-Y$ and 3 vertices with $G-R$, or you could think of it as 2 vertices of B-R and 2 vertices of G-Y. In the latter case, the $2 \mathrm{G}$ ‘s are not consecutive colors of the same chain.
• As you combine more triangles together (the quadrilateral only combined two) and consider the possible colorings, you will see sections of Kempe

chains émerge. Wé will seé hów thése Kémpé chảins èmèrgé in Chápter $21 .$
It’s also easy to see how one color pair (like R-Y) will border its counterpart (B-G):

• Draw one $\mathrm{R}$ vertex and one $Y$ vertex connected by an edge.
• If a new vertex connects to each of these, it must be $B$ or $G$.
• If a new vertex connects to the R but not the $Y$, it may be $Y, B$, or $G$.
• If a new vertex connects to the $Y$ but not the $\mathrm{R}$, it may be $\mathrm{R}, \mathrm{B}$, or $\mathrm{G}$.
• Either the R-Y chain will continue to grow, or it will get surrounded by B and $\mathrm{G}$.
• If you focus on the B and G, you will draw a similar conclusion for its chain.
• If one chain becomes completely surrounded by its counterpart, a new section of the chain may emerge on the other side of its counterpart.
Kempe demonstrated that all vertices with degree four (those which are connected to exactly four other vertices) are four-colorable [Ref. 2]. For example, consider the center vertex below.

数学代写|图论作业代写Graph Theory代考|In the previous figure

In the previous figure, vertex $E$ is degree four since it is connected to four other vertices. Kempe showed that vertices A, B, C, and D can’t be forced to be four different colors, such that vertex E can always be colored without violating the four-color theorem, regardless of how the rest of the MPG may look outside of the portion shown on the previous page.

• A and C are either part of the same section of an A-C Kempe chain, or they each lie on separated sections of A-C Kempe chains. (If $A$ and $C$ are red and yellow, for example, then an A-C chain is a red-yellow chain.) – If $A$ and $C$ each lie on separated sections of an A-C Kempe chain, the colors of one of the sections could be reversed, which effectively recolors C to match A’s color.If A and C are part of the same section of an A-C Kempe chain, B and D must each lie on separated sections of B-D Kempe chains because the A-C Kempe chain will block any B-D Kempe chain from reaching D from B. (If $\mathrm{B}$ and $\mathbf{D}$ are blue and green, for example, then $\mathrm{a}$ B-D Kempe chain is a blue-green chain.) Since B and D each lie on separated sections of B-D Kempe chains in this case, the colors of one of the sections of B-D Kempe chains could be reversed, which effectively recolors D to match B’s color. – Therefore, either C can be made to have the same color as A or D can be made to have the same color as $B$ by reversing a separated section of a Kempe chain.

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

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