On the crossing number of join product of the discrete graph with special graphs of order five

The main aim of this paper is to give the crossing number of the join product G⇤ + Dn for the disconnected graph G⇤ of order five consisting of the complete graph K4 and of one isolated vertex, and where Dn consists of n isolated vertices. In the proofs, the idea of a minimum number of crossings between two different subgraphs by which the graph G⇤ is crossed exactly once will be extended. All methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations. Finally, by adding new edges to the graph G⇤, we are able to obtain the crossing numbers of Gi +Dn for two other graphs Gi of order five.


Introduction
Over the last years, some results concerning crossing numbers of join products of two graphs have been obtained. It is well known that the problem of reducing the number of crossings on the edges in the drawings of graphs was studied in many areas, and the most prominent area is VLSI technology. The lower bound on the chip area is determined by the crossing number and by the number of vertices of the graph. By Garey and Johnson [4] we already know that the computing of the crossing number of a given graph in general is NP-complete problem.

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On the crossing number of join product of the discrete graph ...

| Michal Staš
The crossing number cr(G) of a simple graph G with the vertex set V (G) and the edge set E(G) is the minimum possible number of edge crossings in a drawing of G in the plane. (For the definition of a drawing see [9].) It is easy to see that a drawing with minimum number of crossings (an optimal drawing) is always a good drawing, meaning that no edge crosses itself, no two edges cross more than once, and no two edges incident with the same vertex cross. Let D (D(G)) be a good drawing of the graph G. We denote the number of crossings in D by cr D (G). Let G i and G j be edge-disjoint subgraphs of G. We denote the number of crossings between edges of G i and edges of G j by cr D (G i , G j ), and the number of crossings among edges of G i in D by cr D (G i ). It is easy to see that for any three mutually edge-disjoint subgraphs G i , G j , and G k of G, the following equations hold: In the paper, some proofs will be also based on the Kleitman's result on crossing numbers of the complete bipartite graphs [7]. More precisely, he proved that The exact values for the crossing numbers of G + D n for all graphs G of order at most four are given by Klešč and Schrötter [12]. Also, the crossing numbers of the graphs G + D n are known for few graphs G of order five and six, see [3], [8], [9], [10], [11], and [16]. In all these cases, the graph G is connected and contains at least one cycle. The crossing numbers of the join product G + D n are known only for some disconnected graphs G, and so the purpose of this article is to extend the known results concerning this topic to new disconnected graphs, see [2] and [15].
The methods used in the paper are new, and they are based on combinatorial properties of the cyclic permutations. In [2] and [3] by Berežný and Staš, the properties of cyclic permutations are also verified by the help of software. Also in this article, some parts of proofs can be simplified by utilizing the work of the software COGA that generates all cyclic permutations by Berežný and Buša [1]. The similar methods were partially used earlier in the papers [6] and [14]. We were unable to determine the crossing number of the join product G ⇤ +D n using the methods used in [9], [11], and [12]. Let G ⇤ be the disconnected graph of order five consisting of one isolated vertex and of the complete graph K 4 , and let V (G ⇤ ) = {v 1 , v 2 , . . . , v 5 }. We consider the join product of G ⇤ with the discrete graph on n vertices denoted by D n Clearly, the graph G ⇤ + D n consists of one copy of the graph G ⇤ and of n vertices t 1 , t 2 , . . . , t n , where any vertex t i , i = 1, 2, . . . , n, is adjacent to every vertex of G ⇤ . Let T i , i = 1, . . . , n, denote the subgraph induced by the five edges incident with the vertex t i . This means that the graph T 1 [ · · · [ T n is isomorphic with the complete bipartite graph K 5,n and therefore, we can write

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On the crossing number of join product of the discrete graph ... | Michal Staš 2. Cyclic permutations, configurations, and possible drawings of G ⇤ Let D be a good drawing of the graph G ⇤ + D n . The rotation rot D (t i ) of a vertex t i in the drawing D is the cyclic permutation that records the (cyclic) counter-clockwise order in which the edges leave t i , see [6]. We use the notation (12345) if the counter-clockwise order the edges incident with the vertex t i is t i v 1 , t i v 2 , t i v 3 , t i v 4 , and t i v 5 . We emphasize that a rotation is a cyclic permutation; that is, (12345), (23451), (34512), (45123), and (51234) denote the same rotation. Thus, 5!/5 = 24 different rot D (t i ) can appear in a drawing of the graph G ⇤ + D n . By rot D (t i ) we understand the inverse permutation of rot D (t i ). In the given drawing D, we separate all subgraphs T i , i = 1, . . . , n, of the graph G ⇤ + D n into three mutually disjoint subsets depending on how many times the considered T i crosses the edges of G ⇤ in D. For i = 1, . . . , n, Every other subgraph T i crosses the edges of G ⇤ at least twice in D. Due to arguments in the proof of Theorem 3.1, at least one of the sets R D and S D must be nonempty in a good drawing D of G ⇤ + D n with the smallest number of crossings. For Let us discuss all possible drawings of G ⇤ . Since the graph G ⇤ contains K 4 as a subgraph (for brevity, we write K 4 (G ⇤ )), we only need to consider possibilities of crossings among edges of K 4 (G ⇤ ). If we suppose a good subdrawing of G ⇤ in which the edges of K 4 (G ⇤ ) do not cross each other, then the isolated vertex of G ⇤ can be placed in arbitrary triangular region of D(K 4 (G ⇤ )) and we always obtain the same drawing with respect to isomorphisms that is shown in Figure 1(a). If the edges of K 4 (G ⇤ ) cross each other, then there are next two possibilities depending on in which region of D(K 4 (G ⇤ )) the isolated vertex of G ⇤ is placed and they are shown in Figure 1(b), and (c). The vertex notation of the graph G ⇤ in Figure 1 will be justified later. First, let us assume a good drawing D of the graph G ⇤ + D n in which the edges of G ⇤ do not cross each other. In this case, without loss of generality, we can choose the vertex notation of the graph G ⇤ in such a way as shown in Figure 1(a). It is obvious that the set R D is empty, and so our aim is to list all possible rotations rot D (t i ) which can appear in D if the edges of T i cross the edges of G ⇤ exactly once. Since there is only one subdrawing of F i \ {v 3 , v 5 } represented by the rotation (142), there are three possibilities how to obtain the subdrawing of F i \ v 5 depending on which edge of the graph G ⇤ is crossed by the edge t i v 3 . Every of these three subdrawings of F i \ v 5 produces four drawings of F i depending on in which region the edge t i v 5 is placed. We denote these twelve possibilities under our consideration by A k , and B k , for k = 1, . . . , 6. The configuration is of type A or B, if the vertex v 5 is placed in the triangular region with two vertices or with one vertex of G ⇤ on its boundary in the subdrawing D(F i \ v 5 ), respectively. In the rest of the paper, each cyclic permutation is represented by the permutation with 1 in the first position. Thus, the configurations Now, we deal with the minimum numbers of crossings between two different subgraphs T i and T j depending on the configurations of subgraphs F i and F j . Let D be a good drawing of the graph G ⇤ + D n , and let X , Y be configurations from M D . We shortly denote by cr D (X , Y) the number of crossings in D between } over all pairs X and Y from M among all good drawings of the graph G ⇤ + D n . Our aim is to establish cr(X , Y) for all pairs X , Y 2 M. In particular, the configurations A 1 and A 2 are represented by the cyclic permutations (14325) and (14523), respectively. Since the minimum number of interchanges of adjacent elements of (14325) required to produce cyclic permutation (14523) = (13254) is one, any subgraph T j with the configuration A 2 of F j crosses the edges of T i with the configuration A 1 of F i at least once 1 , i.e., cr(A 1 , A 2 ) 1. The same reason gives cr( Moreover, the Woodall's result for m = 5 also implies that cr(A p , B p ) 4 holds for any p = 1, . . . , 6, and cr(B p , B q ) 4 holds with respect to the restrictions p ⌘ q (mod 2), where p, q = 1, . . . , 6. Clearly, also cr(A p , A p ) 4 for any p = 1, . . . , 6. For all remaining pairs of configurations are established the minimum numbers of crossings at least three. For any T i 2 S D with the configuration B 1 of F i , if there is a subgraph T j 2 S D , j 6 = i such that cr D (T i , T j )  2, then the vertex t j must be placed in the triangular region with two vertices v 1 , v 2 or in the quadrangular region with two vertices v 1 , v 5 of G ⇤ on its boundary in the subdrawing D(F i ). Hence, the subgraph F j is exactly represented by rot D (t j ) = (15423) or rot D (t j ) = (13452), and therefore, cr(B 1 , A p ) 3 and cr(B 1 , B p ) 3 hold for each p = 2, 4, 6. Similar arguments can be applied for the configurations B q of some subgraph F i for q = 2, . . . , 6. The resulting lower bounds for the number of crossings of configurations from M are summarized in the symmetric Table 1. (Here, X p and Y q are configurations of the subgraphs F i and F j , where p, q 2 {1, . . . , 6} and X , Y 2 {A, B}.) Assume a good drawing D of the graph G ⇤ + D n in which the edges of G ⇤ cross each other exactly once and the isolated vertex of the graph G ⇤ is placed in the quadrangular region in the www.ejgta.org On the crossing number of join product of the discrete graph ... | Michal Staš 1  4  1  2  3  2  3  4  3  2  3  2  3  A 2  1  4  3  2  3  2  3  4  3  2  3  2  A 3  2  3  4  1  2  3  2  3  4  3  2  subdrawing D(G ⇤ \ v 5 ). In this case, without loss of generality, we can choose the vertex notation of the graph G ⇤ in such a way as shown in Figure 1 Table 2. (Here, E p and E q are configurations of the subgraphs F i and F j , where p, q 2 {1, 2, 3, 4}.) 3. The crossing number of G ⇤ + D n Two vertices t i and t j of the graph G ⇤ + D n are antipodal in a drawing of G ⇤ + D n if the subgraphs T i and T j do not cross. A drawing is antipode-free if it has no antipodal vertices. Now we are able to prove the main result of this paper. Proof. In Figure 3 there is the drawing of the graph . We prove the reverse inequality by induction on n. The graph G ⇤ + D 1 contains K 5 as a subgraph and the graph G ⇤ + D 2 is a subdivision of K 6 . It was proved in [5] that cr(K 5 ) = 1 and cr(K 6 ) = 3. So, the result is true for n = 1 and n = 2. Suppose now that, for some n 3, there is a drawing D with and that for any positive integer m < n. (2) www.ejgta.org On the crossing number of join product of the discrete graph ...

| Michal Staš
Our assumption on D together with cr(K 5,n ) = 4 ⌅ n 2 ⇧⌅ n 1 2 ⇧ implies that Moreover, if r = |R D | and the set S D is empty, then which forces r In the case, if s = |S D | and the set R D is empty, then which implies s ⌃ n 2 ⌥ + 1 + cr D (G ⇤ ). Now, for T i 2 R D [ S D , we discuss the existence of possible configurations of subgraph F i = G ⇤ [ T i in the drawing D and we show that in all cases a contradiction with the assumption (1) is obtained.
Case 1: cr D (G ⇤ ) = 0. Without loss of generality, we can choose the vertex notation of the graph G ⇤ in such a way as shown in Figure 1(a). Since the set R D is empty, we deal with the configurations belonging to the nonempty set M D according to inequality (4).
We claim that the considered drawing D must be antipode-free. Of course, if T k and T l are two different subgraphs from the nonempty set S D , then the vertices v k and v l are not antipodal due to the positive values in Table 1. For a contradiction, suppose that cr D (T k , T l ) = 0, and at least one of the subgraphs T k and T l is not included in the set S D , which yields that cr D (G ⇤ , T k [ T l ) 3. This contradiction with the assumption (1) confirms that D is antipode-free. For T i 2 S D , we deal with the configurations belonging to the set M D and we discuss over all possible subsets of M D in the following subcases: Without lost of generality, let us consider two different subgraphs T n 1 , T n 2 S D such that F n 1 and F n have configurations A 1 and A 2 , respectively. Then, cr D (T n 1 [ T n , T k ) 5 is fulfilling for any T k 2 S D with k 6 = n 1, n by summing the values in all columns in the first two rows of Table 1. Moreover, cr D (T n 1 [ T n , T k ) 3 holds for any T k 6 2 S D provided by the minimum number of interchanges of adjacent elements of rot D (t n 1 ) required to produce the cyclic permutation rot D (t n ) is three. As cr D (G ⇤ [ T n 1 [ T n ) 3, by fixing the graph T n 1 [ T n , we have This contradicts the assumption of D. Due to the symmetry, the same arguments are applied for the cases {A 3 , A 4 } and {A 5 , Without lost of generality, let us consider three different subgraphs T n 2 , T n 1 , T n 2 S D such that F n 2 , F n 1 and F n have configurations A 1 , A 3 and B 5 , respectively. Then, cr D (T n 2 [ T n 1 [ T n , T k ) 8 holds for any T k 2 S D with k 6 = n 2, n 1, n by summing of three corresponding values of Table 1. Moreover, if there is a subgraph T k , k 6 = n 1, n such that cr D (T n 1 [T n , T k ) = 2, then the minimum number of interchanges of adjacent elements of rot D (t n 2 ) required to produce the inverse cyclic permutation of rot D (t k ) is at least two, and so cr D (T n 2 [T n 1 [T n , T k ) 4 holds for any T k 6 2 S D . As cr D (T n 2 [T n 1 [T n ) 6, by fixing the graph T n 2 [T n 1 [T n , we have This also contradicts the assumption of D. The verification for all seven other possibilities proceeds in the same way and therefore, in the next part, suppose that {A p , A p+2 , A p+4 } 6 ✓ M D for any p = 1, 2, and also {A o , A p , B q } 6 ✓ M D with o ⌘ p ⌘ q (mod 2) for any three mutually different o, p, q = 1, . . . , 6. Now, for T i 2 S D , we will discuss the possibility of obtaining a subdrawing of G ⇤ [ T i [ T j in D with cr D (T i , T j ) = 2 for some T j 2 S D . Let us consider that there are two subgraphs T i , T j 2 S D with cr D (T i , T j ) = 2 such that F i and F j have configurations X p and Y q , respectively, where X , Y 2 {A, B} and p, q 2 {1, . . . , 6}. Then, cr D (T i [ T j , T k ) 6 holds for any T k 2 S D , k 6 = i, j by summing of two corresponding values of Table 1