Calculating Crossing Numbers of Graphs Using Combinatorial Principles

e crossing numbers of graphs were started fromTurán’s brick factory problem (TBFP). Because of its wide range of applications, it has been used in computer networks, electrical circuits, and biological engineering. Recently, many experts began to pay much attention to the crossing number of G\e, which obtained from graph G by deleting an edge e. In this paper, by using some combinatorial skills, we determine the exact value of crossing numbers ofK1,4,n\e andK2,3,n\e. ese results are an in-depth work of TBFP, which will be benecial to the further study of crossing numbers and its applications.


Introduction
e concept of crossing number was introduced by the Hungarian mathematician Turán [1]. He encountered a practical problem in Budapest brick factory, which named "Turán's brick factory problem"(TBFP). In fact, TBFP is to determine the minimal number of crossings among edges of the complete bipartite graph K m,n .
In the past ve decades, it turned out that the crossing numbers have strong practical signi cance. And they can be widely used in various elds, such as the VLSI circuit layout [2], the identi cation and repaint of sketch [3], and automatic generation of ER diagram in software development [4] (see [5,6]). One of important applications is to nd the best location for a new electrical substation so that every two substations are directly connected and to do without overlapping power lines. Rach [7] has applied the crossing number to solve a problem of locating electrical substations in the city of Glencoe, MN.
With the depth of research, the crossing numbers of graphs have been investigated extensively in the mathematical, computer, and biological literature, often under di erent parameters, such as the parity [8], odd-crossing number [9], regular graphs [10], chromatic number [11], and genus [12]. For more results and its properties about crossing numbers, reader can refer to [13][14][15][16]. As for the complete bipartite graphs K m,n , Kleitman in [17] proved that cr K m,n m 2 m − 1 2 Recently, the crossing numbers of complete multipartite graphs attracted much attention. In 1986, Asano [18] obtained the crossing numbers of graphs K 1,3,n and K 2,3,n . In 2008, Huang and Zhao in paper [19] established that the crossing number of graph K 1,4,n is equal to n(n − 1). In 2020, the crossing numbers of graphs K 1,1,4,n as well as K 1,1,4 □T have been proved in [20].
While studying the crossing number of the primal graph G, many experts also began to follow with interest the crossing number of G\e, which is obtained by deleting an edge e from G. is is an interesting problem worthy of consideration. For G, it is a complete graph or complete bipartite graph. Ouyang in [21,22] as well as Chia in [23], independently, established the precise values of crossing numbers of certain graphs G\e: (1) K n \e for n ≤ 12, (2) K m,n \e for m 3, 4 and n ≥ 1.
Very recently, Huang and Wang in [24] by applying the method of edge-labeling, which is new and di erent from those used in [22,23], have proved that cr(K m,n \e) � (n − 1)(n − 2). In the present paper, We attempt to study the crossing number of graph G\e, where G is a complete tripartite graph. Using Huang's result and some combinatorial skills, we establish exact value of crossing numbers of K 1,4,n \e and K 2,3,n \e.
Terms and definitions involved in the paper follow as those in [24]. Given a simple graph G, the crossing number denoted by cr(G) is the minimum number of crossings in any good drawing of graph G.
e good drawing with minimum number of crossings is called the optimal drawing. For a vertex v ∈ V(G), E v is used to represent all edges which is incident with v. e responsibilities of v, denoted r D (v), are defined to be the sum of crossings of all edges incident to v in the drawing D. 4 , and Z � z 1 , z 2 , . . . , z n . Let XY, XZ, and YZ be the subgraphs of K 1,4,n induced by X ∪ Y, X ∪ Z, and Y ∪ Z, respectively. Clearly, XY � K 1,4 . For i � 1, 2, . . . , n, let E z i be the subgraph of K 1,4,n induced by five edges incident with the vertex z i . We will easily get the following formula:

Lemma 4.
For an edge e in complete tripartite graph K 1,4,n , then Proof. Let D be an optimal drawing of K 5,n having Z(5, n) crossings due to Zarankiewicz [17]. In the drawing, we place the vertices of K 5,n at coordinators (0, i) and (j, 0) where − 2 ≤ i ≤ 3, − ⌊n/2⌋ ≤ j ≤ ⌈n/2⌉, and i, j ≠ 0. en, we join (0, i) to (j, 0) with straight line segment. Next, we join the edges of K 1,4 � xy i , 1 ≤ i ≤ 4 as shown in Figure 1, and an optimal drawing of the complete tripartite graph K 1,4,n is obtained. Let us denote the drawing by D ′ . It is not difficult to see that cr D′ (K 1,4,n ) � cr(K 1,4,n ) � Z(5, n) + 2⌊n/2⌋. In the following, we obtain the graph K 1,4,n \e together with its drawing from D ′ .
(i) If e ∈ XY. By deleting the edge e � xy 1 from D ′ , then a drawing of K 1,4,n \e xy is obtained. We can easily check that there are ⌊n/2⌋ crossings on the edge of xy 1 . erefore, we can verify that (ii) If e ∈ XZ. en, by deleting the edge of e � xz ⌈n/2⌉ , we can obtain a drawing of K 1,4,n \e xz . Likewise, the responsibility of the edge xz ⌈n/2⌉ is 2(⌈n/2⌉ − 1). So, we can obtain that In the optimal drawing of K 5,n , which have Z(5, n) crossings given by Zarankiewicz, we reconnect the edges of K 1,4 � xy i , 1 ≤ i ≤ 4 , as shown in Figure 2. And then, by deleting the edge of e � y 1 z 1 , a drawing of K 1,4,n \e yz is obtained. Let us denote the drawing by D ′ . en, one can verify that the responsibility of the edges K 1,4 and y 1 z 1 is n and 2(⌈n/2⌉ − 1) + 1, respectively. erefore, we have Combined with the above three cases, this completes the proof. □ Theorem 1. For any edge e xy in complete tripartite graph K 1,4,n , where e xy ∈ XY, then Proof. It is not difficult to know that K 1,4,n \e xy contains a subgraph that is isomorphic to (S 3 ∪ K 1 ) + nK 1 . And it was shown from Lemma 3 that cr((S 3 ∪ K 1 ) + nK 1 ) � Z(5, n) +⌊n/2⌋. us, cr(K 1,4,n \e xy ) ≥ cr((S 3 ∪ K 1 ) + nK 1 ) � Z(5, n) +⌊n/2⌋. e reverse inequalities are confirmed by Lemma 4. is completes the proof.

Mathematical Problems in Engineering
Finally, combining with inductive hypothesis and Lemma 1, we have that us, the proof of eorem 2 is nished. □ Theorem 3. For any edge e xz in complete tripartite graph K 1,4,n , where e xz ∈ XZ, then Proof.
Without losing generality, we assume that under the drawing φ of K 1,4,n \e xz , the clockwise order of these four 4 ). And we assume e xz xz 1 . us, the graph K 1,4,n \e xz has an additional (n − 1) edges 4 denote the sets of all those images xz i , each of which places in the angle α i is formed between φ(xy i ) and φ(xy i+1 ), where the indices are read modulo 4 (see Figure 3(a)). We note that |A 1 | + |A 2 | + |A 3 | + |A 4 | n − 1. Further more, we see that in the plane R 2 , there exists a circular neighborhood around where ε is a positive number small enough such that for any other edge y i z j (i 1, 2, 3, 4; j 1, 2, . . . , n) of K 1,4,n \e xz not incident with x, φ(y i z j ) ∩ N(φ(x), ε) ∅. Next, we divide two cases to discuss. □ Case 1. assume (n − 1) is odd. us, we have |A 1 | ≠ |A 3 | or |A 2 | ≠ |A 4 |. Otherwise, n − 1 4 I 1 |A i | 2(|A 1 | + |A 2 |); this contradicts the fact that (n − 1) is odd. Without loss of generality, we assume |A 1 | ≠ |A 3 |, and more precisely, let |A 3 | ≥ |A 1 | + 1. In the following, we produce the graph K 5,n+1 \e together with its drawing φ' by three steps.

Corollary 1.
For an edge e in complete tripartite graph K 1,4,n , then
(i) If e ∈ XY. By deleting the edge e � x 2 y 2 from D ′ , then a drawing of K 2,3,n \e xy is obtained. It is easily checked that there are ⌈n/2⌉ crossings on the edge of x 2 y 2 . erefore, we can verify that cr K 2,3,n \e xy ≤ Z(5, n) + n − ⌈ n 2 � Z(5, n) +⌊ n 2 . (22) (ii) If e ∈ XZ. We can get a drawing of K 2,3,n \e xz by deleting the edge e � x 1 z ⌈ n/2 from D ′ . Obviously, the responsibility of the edge of x 1 z ⌈n/2⌉ is 2(⌈n/2⌉ − 1). So, we have (iii) If e ∈ YZ. e drawing of K 2,3,n \e yz can be obtained by deleting the edge e � y 1 z 1 from D ′ . We note that the edge y 1 z 1 crosses with the edges 〈YZ ∪ x 2 y 2 〉 exactly 2(⌈n/2⌉ − 1) + 1 times. us, we obtain that erefore, according to the above analysis, we have completed the proof.
Proof. Without loss of generality, let e xz � x 1 z n . erefore, according to (4), we note that At first, we can obtain by Lemma 7 that cr(K 2,3,n \e xz ) ≤ n 2 − 2n + 2. It is not difficult to know that K 2,3,1 \e xz contains a subgraph which is isomorphic to K 3,3 and K 2,3,2 \e xz contains a subgraph which is isomorphic to K 3,4 . us, we have cr(K 2,3,1 \e xz ) ≥ 1 and cr(K 2,3,2 \e xz ) ≥ 2, so the theorem is established for n � 1 and 2. Now, we suppose that n ≥ 3 and that cr(K 2,3,k \e xz ) ≤ k 2 − 2k + 2 for any 3 ≤ k ≤ n − 1. We will derive contradiction to prove the Mathematical Problems in Engineering reverse inequality. We assume to contrary that K 2,3,n /e xz has a good drawing D such that cr D K 2,3,n \e xz ≤ n 2 − 2n + 1.
In the subsequent proof process, we always deduce some contradictions to ( * ). We rst have the following claims.

Proof.
Otherwise, we have cr D (K 2,3 ) + cr D (K 2,3 , ∪ n i 1 E z i ) ≥ n. As ∪ n i 1 E z i is isomorphic to K 5,n \e, and it was proved by Lemma 2 that cr(K 5,n \e) (n − 1)(n − 2). us, we obtain cr D K 2,3,n \e xz cr D K 2,3 ∪ ∪ is also contradicts to ( * ). Furthermore, we have the following claim.
Proof. If cr D (K 2,3 ) 0, then the good drawing of K 2,3 induced by D divides the plane into three quadrangular regions ω(y 1 , y 2 ), ω(y 2 , y 3 ), and ω(y 3 , y 1 ) depending on which two of the vertices y 1 , y 2 , y 3 , and y 4 are placed on the corresponding boundary. us, under the drawing of K 2,3 , we have cr D (K 2,3 , E z n ) ≥ 1 and cr D (K 2,3 , E z i ) ≥ 1 for all i 1, 2, . . . , n − 1. In other words, the edges of K 2,3 are crossed at least n times by the subgraphs ∪ n− 1 i 1 E z i ; this contradicts with Claim 2.   Figure 4: An optimal drawing of K 2,3,n . 6 Mathematical Problems in Engineering If cr D (K 2,3 ) ≥ 1, combining together with Claim 2, we can obtain that cr D (K 2,3 , ∪ n− 1 i�1 E z i ) ≤ n − 2. is forces that there exists a vertex z i (1 ≤ i ≤ n − 1) such that cr D (K 2,3 , E z i ) � 0. us, Claim 3 is proved. Now, we continue to prove the theorem. By Claim 3, without loss of generality, we assume cr D (K 2,3 , E z 1 ) � 0.
us, there is a disk such that the five vertices of K 2,3 are all placed on the boundary of disk. We assume the vertex z 1 placed in the external of the disk, and the edges of K 2,3 are all placed in the inner side of the disk. It is easy to obtain that two drawing of 〈K 2,3 ∪ E z 1 〉 as shown in Figure 5.
In Figure 5, except for the region which marked with α, each of regions contains at most two vertices of K 2,3 in its boundary. Hence, we obtain that cr D (K 2,3 ∪ E z 1 , E z i ) ≥ 3. When z i is placed in the region α, we have cr D (K 2,3 ∪ E z 1 , E z i ) ≥ 2, if and only if cr D (K 2,3 , E z i ) � 2 and the equality cr D (E z 1 , E z i ) � 0 holds. is together with Claim 1 implies that cr D (K 2,3 ∪ E z 1 , E z i ) ≥ 3.
Moreover, each region contains at most three vertices of K 2,3 in its boundary in Figure 5. us, erefore, it follows from cr D (K 2,3 ∪ E z 1 ) � 1, 〈 ∪ n− 1 i�1 E z i 〉 � K 5,n \e, and the assumption of the theorem that Clearly, this contradicts to ( * ). In summary, the hypothesis is not true, and the proof is done.
Together with eorems 4-6, we can get the following corollary immediately.

Conclusion
e problem crossing numbers of graphs are originated in a practical application, whose theory has been widely applied in many fields. However, determining the crossing numbers of graphs are NP-complete. Because of its difficulty, the research progress is slow. In this paper, according to the structural characteristics of complete multipartite graph, using "drawing restriction method," "embedding method," and "point degree local modification method," we determine the exact value of crossing numbers of K 1,4,n \e and K 2,3,n \e. ese results are an in-depth work of TBFP, which will be beneficial to the further study of crossing numbers and its applications.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.
Mathematical Problems in Engineering 7