Abstract
The crossing numbers of graphs were started from Turán’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 , which obtained from graph by deleting an edge . In this paper, by using some combinatorial skills, we determine the exact value of crossing numbers of and . These results are an in-depth work of TBFP, which will be beneficial to the further study of crossing numbers and its applications.
1. Introduction
The 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 .
In the past five decades, it turned out that the crossing numbers have strong practical significance. And they can be widely used in various fields, such as the VLSI circuit layout [2], the identification and repaint of sketch [3], and automatic generation of ER diagram in software development [4] (see [5, 6]). One of important applications is to find 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 different 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 [13–16]. As for the complete bipartite graphs , Kleitman in [17] proved that
Recently, the crossing numbers of complete multipartite graphs attracted much attention. In 1986, Asano [18] obtained the crossing numbers of graphs and . In 2008, Huang and Zhao in paper [19] established that the crossing number of graph is equal to . In 2020, the crossing numbers of graphs as well as have been proved in [20].
While studying the crossing number of the primal graph , many experts also began to follow with interest the crossing number of , which is obtained by deleting an edge from . This is an interesting problem worthy of consideration. For , 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 : (1) for , (2) for and .
Very recently, Huang and Wang in [24] by applying the method of edge-labeling, which is new and different from those used in [22, 23], have proved that . In the present paper, We attempt to study the crossing number of graph , where is a complete tripartite graph. Using Huang’s result and some combinatorial skills, we establish exact value of crossing numbers of and .
Terms and definitions involved in the paper follow as those in [24]. Given a simple graph , the crossing number denoted by is the minimum number of crossings in any good drawing of graph . The good drawing with minimum number of crossings is called the optimal drawing. For a vertex , is used to represent all edges which is incident with . The responsibilities of , denoted , are defined to be the sum of crossings of all edges incident to in the drawing .
2. Crossing Number of
Let be the vertex partition of the graph , where , , and . Let , , and be the subgraphs of induced by , , and , respectively. Clearly, . For , let be the subgraph of induced by five edges incident with the vertex . We will easily get the following formula:
Lemma 1 (see [19]). for any .
Lemma 2 (see [24]). for any .
Lemma 3 (see [13, 25]). for any .
Lemma 4. For an edge in complete tripartite graph , then
Proof. Let be an optimal drawing of having crossings due to Zarankiewicz [17]. In the drawing, we place the vertices of at coordinators and where , , and . Then, we join to with straight line segment.
Next, we join the edges of as shown in Figure 1, and an optimal drawing of the complete tripartite graph is obtained. Let us denote the drawing by . It is not difficult to see that . In the following, we obtain the graph together with its drawing from .(i)If . By deleting the edge from , then a drawing of is obtained. We can easily check that there are crossings on the edge of . Therefore, we can verify that(ii)If . Then, by deleting the edge of , we can obtain a drawing of . Likewise, the responsibility of the edge is . So, we can obtain that(iii)If . In the optimal drawing of , which have crossings given by Zarankiewicz, we reconnect the edges of , as shown in Figure 2. And then, by deleting the edge of , a drawing of is obtained. Let us denote the drawing by . Then, one can verify that the responsibility of the edges and is and , respectively. Therefore, we haveCombined with the above three cases, this completes the proof.
Theorem 1. For any edge in complete tripartite graph , where , then
Proof. It is not difficult to know that contains a subgraph that is isomorphic to . And it was shown from Lemma 3 that . Thus, . The reverse inequalities are confirmed by Lemma 4. This completes the proof.
Theorem 2. For any edge in complete tripartite graph , where , then
Proof. At first, according to Lemma 4, we have , and Theorem 2 is true if the equality holds. For since is planar graph, therefore, is true. Now, we suppose that for any positive integer .
Let be an optimal drawing of graph , which satisfies . Without loss of generality, we say . By deleting the edges of from drawing , the graph is obtained. Hence, we get thatLikewise, for any , we have the following thatOtherwise, by deleting the edges of for any , we obtain the graph which is isomorphic to and has less than crossings. Therefore, from (9) and (10), summing up for , we can obtain thatThus, we simplify and conclude thatFinally, combining with inductive hypothesis and Lemma 1, we have thatThus, the proof of Theorem 2 is finished.
Theorem 3. For any edge in complete tripartite graph , where , then
Proof. From Lemma 4, we can get that . Thus, Theorem 3 is true; we need only to prove that for any drawing of .
Without losing generality, we assume that under the drawing of , the clockwise order of these four images around is . And we assume . Thus, the graph has an additional edges incident with . Let denote the sets of all those images , each of which places in the angle is formed between and , where the indices are read modulo (see Figure 3(a)). We note that . Further more, we see that in the plane , there exists a circular neighborhood around such that , where is a positive number small enough such that for any other edge of not incident with , . Next, we divide two cases to discuss.
Case 1. assume is odd. Thus, we have or . Otherwise, ; this contradicts the fact that is odd. Without loss of generality, we assume , and more precisely, let . In the following, we produce the graph together with its drawing by three steps.
Step 1. Add a new vertex in some location of .
Step 2. For all , delete the partition of lying in (do not delete the vertex ).
Step 3. Connect to each vertex in in such a way as described in Figure 3(b).
Thus, we obtain a drawing of the graph from the drawing of . It is easy to obtain thatWith the help of by Lemma 2, we have that
Case 2. assume is even. We consider arbitrarily a pair of numbers and or and . Without losing generality, we say and , again we say . Completely analogously to Case 1 above, we can get a drawing of graph such thatNow, applying , we obtain the following thatThus, by the arguments derived in Cases 1 and 2, we have shown that for any drawing of . This completes the proof.
Therefore, from Theorems 1–3, we can get the following corollary immediately.
Corollary 1. For an edge in complete tripartite graph , then
3. Crossing Number of
In graph , let be the vertex partition, where , , . We denote , , be the subgraphs of induced by respectively. Clearly, . For let be the subgraph of induced by five edges incident with . We can easily get that
Lemma 5 (see [18]). for any .
Lemma 6 (see [26]). for any .
Lemma 7. For an edge in complete tripartite graph , then
Proof. Let be an optimal drawing of having crossings due to Zarankiewicz [17]. In the drawing, we place the vertices of at coordinators and where and . Then, we join to with straight line segment.
Next, we join the edges of , as shown in Figure 4, and thus an optimal drawing of the complete tripartite graph is obtained. Let us denote the drawing by . We can easily check that . In the following, we obtain the graph together with its drawing from .(i)If . By deleting the edge from , then a drawing of is obtained. It is easily checked that there are crossings on the edge of . Therefore, we can verify that (ii)If . We can get a drawing of by deleting the edge from . Obviously, the responsibility of the edge of is . So, we have(iii)If . The drawing of can be obtained by deleting the edge from . We note that the edge crosses with the edges exactly times. Thus, we obtain thatTherefore, according to the above analysis, we have completed the proof.
Theorem 4. For any edge in complete tripartite graph , where , then
Proof. It is not difficult to know that is isomorphic to . And it was shown by Lemma 6 that . Thus, . The reverse inequalities are confirmed by Lemma 7. Hence, the proof is done.
Theorem 5. For any edge in complete tripartite graph , where , then
Proof. Without loss of generality, let . Therefore, according to (4), we note thatAt first, we can obtain by Lemma 7 that . It is not difficult to know that contains a subgraph which is isomorphic to and contains a subgraph which is isomorphic to . Thus, we have and , so the theorem is established for and . Now, we suppose that and that for any . We will derive contradiction to prove the reverse inequality. We assume to contrary that has a good drawing such thatIn the subsequent proof process, we always deduce some contradictions to . We first have the following claims.
Claim 1. for all , and .
Proof. Otherwise, without loss of generality, let . According to Lemma 5, implies that . As isomorphic to , and it was shown by Lemma 2 that . Thus, . The known fact that implies that for all . Therefore, we haveClearly, this contradicts to .
Claim 2. .
Proof. Otherwise, we have . As is isomorphic to , and it was proved by Lemma 2 that . Thus, we obtainThis also contradicts to .
Furthermore, we have the following claim.
Claim 3. There exists a vertex such that .
Proof. If , then the good drawing of induced by divides the plane into three quadrangular regions , , and depending on which two of the vertices , and are placed on the corresponding boundary. Thus, under the drawing of , we have and for all . In other words, the edges of are crossed at least times by the subgraphs ; this contradicts with Claim 2.
If , combining together with Claim 2, we can obtain that . This forces that there exists a vertex such that . Thus, Claim 3 is proved.
Now, we continue to prove the theorem. By Claim 3, without loss of generality, we assume . Thus, there is a disk such that the five vertices of are all placed on the boundary of disk. We assume the vertex placed in the external of the disk, and the edges of are all placed in the inner side of the disk. It is easy to obtain that two drawing of as shown in Figure 5.
In Figure 5, except for the region which marked with , each of regions contains at most two vertices of in its boundary. Hence, we obtain that . When is placed in the region , we have , if and only if and the equality holds. This together with Claim 1 implies that .
Moreover, each region contains at most three vertices of in its boundary in Figure 5. Thus, . Therefore, it follows from , , and the assumption of the theorem thatClearly, this contradicts to .
In summary, the hypothesis is not true, and the proof is done.
Using the method completely similar to Theorem 5, we can get the following Theorem 6. Thereby, the proof process of Theorem 6 is omitted here.
Theorem 6. For any edge in complete tripartite graph , where , then
Since is isomorphic to , thus we have the following corollary.
Corollary 2.
Together with Theorems 4–6, we can get the following corollary immediately.
Corollary 3. For an edge in complete tripartite graph , then
4. Conclusion
The 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 and . These results are an in-depth work of TBFP, which will be beneficial to the further study of crossing numbers and its applications.
Finally, we give some conjecture and open problems.
Conjecture 1. for
Problem 1. For an edge in complete tripartite graph , , and . Then what are the precise values of crossing numbers of , , and ?
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the scientific research project of Hunan Provincial Department of Education (2022).