# 数学代写|图论作业代写Graph Theory代考|MS-E1050

#### Doug I. Jones

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## 数学代写|图论作业代写Graph Theory代考|Analyzing Optimal Drawings of G13

We divide the proof of Theorem 2 into two steps. We call red the edges of $G_{13}$ (Fig. 1(a)) which form the path with multiple edges on $\left(u_{5}, u_{4}, u_{3}, u_{2}, u_{1}, x_{1}, x_{2}\right.$, $\left.v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\right)$, and we call blue the edges $\left{u_{i}, v_{j}\right}$ where $i, j \in{1,2,3,4}$.
(1) We will first show that there is an optimal drawing (i.e., one minimizing the number of crossings) of $G_{13}$ such that no red edge crosses a red or a blue edge. Note that blue-blue crossings are still allowed (and likely to occur).
(2) While considering drawings as in the first point, we will focus only on selected crossings (roughly, those involving a blue edge), and prove at least 13 of them, or at least 12 with the remaining drawing still being non-planar.
Proposition 1 (folklore). a) If $D$ is an optimal drawing of a graph $G$, then two edges do not cross more than once, and not at all if sharing a common end.
b) If e and $f$ are parallel edges in $G$ (i.e., $e, f$ have the same end vertices), then there is an optimal drawing of $G$ in which $e$ and $f$ are drawn “closely together”, meaning that they cross the same other edges in the same order.

In view of Proposition 1(b), we adopt the following view of multiple edges: If the vertices $u$ and $v$ are joined by $p$ parallel edges, we view all $p$ of them as one edge $f$ of weight $p$. If (multiple) edges $f$ and $g$ of weights $p$ and $q$ cross each other, then their crossing naturally contributes the amount of $p \cdot q$ to the total number of crossings. With help of the previous, we now finish the first step:
Lemma 1. There exists an optimal drawing of the graph $G_{13}$ in which no red edge crosses a red or a blue edge, or $\operatorname{cr}\left(G_{13}\right) \geq 13$.

## 数学代写|图论作业代写Graph Theory代考|Counting Selected Crossings in a Drawing of G13

In the second step, we introduce two additional sorts of edges of $G_{13}$. The edges $u_{5} x_{1}$ and $v_{5} x_{2}$ are called green, and all remaining edges of $\left({-} x{13}\right.$. I et $C_{-} x_{0}$ denote the subgraph of $G_{13}$ formed by all red and gray edges and the incident vertices. Let $R$ denote the (multi)path of all red edges.

Lemma 2. Let $D$ be an optimal drawing of $G_{13}$ as claimed by Lemma 1. If the subdrawing of $G_{0}$ within $D$ is planar, then $D$ has at least 13 crossings.

Proof (a sketch). There are only two non-equivalent planar drawings of $G_{0}$, as in Fig. 2. We picture them with the red path $R$ drawn as a horizontal line. We call a blue edge of $G_{13}$ bottom if it is attached to $R$ from below at both ends, and top if attached from above at both ends. A blue edge is switching if it is neither top nor bottom. Note that we have only crossings involving a green edge, or crossings of a blue edge with a blue or gray edge.
If $G_{0}$ is drawn as in Fig. 2(a), by the Jordan curve theorem, we deduce:
(I) If a blue edge $e$ is bottom (top), and a blue edge $e^{\prime} \neq e$ attaches to $R$ from below (from above) at its end which is between the ends of $e$ on $R$, then $e$ and $e^{\prime}$ cross. In particular, two bottom (two top) blue edges always cross.
(II) A top (switching) blue edge crosses at least 4 (at least 3 ) gray edges.
(III) If there is weight $k$ of top (or switching) blue edges and weight $\ell$ of bottom blue edges, then each (or at least one) green edge must cross $\min (k, \ell) \leq 3$ blue or red edges.

## 数学代写|图论作业代写Graph Theory代考|Analyzing Optimal Drawings of G13

(1) 我们将首先证明存在一个最优的绘图 (即，一个最小化交叉次数的绘图) $G_{13}$ 使得没 有红边穿过红边或蓝边。请注意，蓝监交叉仍然是允许的（并且可能会发生）。
(2) 在考虑第一点的图纸时，我们将只关注选定的交叉点（大致是那些涉及监色边缘的交 叉点），并证明其中至少有 13 个，或者至少有 12 个，而其余的图形仍然是非平面的. 我们从关于交叉数的一些基本事实开始。

b) 如果 e 和 $f$ 是平行边 $G$ (IE， $e, f$ 具有相同的末端顶点)，那么有一个最佳绘图 $G$ 其中 $e$ 和 $f$ 被“尜密地结合在一起”，这意味着它们以相同的顺序穿过相同的其他边。

## 数学代写|图论作业代写Graph Theory代考|Counting Selected Crossings in a Drawing of G13

(I) 如果一个蓝色边豚 $e$ 是底部 (顶部) 和蓝色边缘 $e^{\prime} \neq e$ 附于 $R$ 从下面 (从上面) 在它的 末端，它在末端之间 $e$ 上 ，然后 $e$ 和 $e^{\prime}$ 叉。特别是，两个底部 (两个顶部) 蓝色边䖶总 是交叉。
(II) 顶部 (切换) 蓝色边傢与至少 4 个 (至少 3 个) 灰色边缘交叉。
（三）如果有重量 $k$ 顶部 (或切换) 蓝色边缘和重量 $\ell$ 底部蓝色边逐，然后每个（或至少一 个) 绿色边豚必须交叉 $\min (k, \ell) \leq 3$ 蓝色或红色边豚。

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