- Email: [email protected]
The Join of Graphs and Crossing Numbers
Electronic Notes in Discrete Mathematics 28 (2007) 349–355 www.elsevier.com/locate/endm
The Join of Graphs and Crossing Numbers Mari´an Kleˇsˇc
1,2
Department of Mathematics Faculty of Electrical Engineering and Informatics Technical University 042 00 Koˇsice, Slovak Republic
Abstract It has been long–conjectured by Zarankiewicz that the crossing number of the comm n−1 n plete bipartite graph Km,n equals m−1 2 2 2 2 . This conjecture has been verified by Kleitman for min{m, n} ≤ 6. Using these results, we give the exact values of crossing numbers for join of two paths, join of two cycles, and for join of path and cycle. In addition, we give the exact values of crossing numbers for join products G + Pn and G + Cn for all graphs G of order at most four. Keywords: graph, join product, drawing, crossing number, path, cycle.
1
Introduction
The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane. It is easy to see that a drawing with the minimum number of crossings must be a good drawing; that is, each two edges have at most one point in common, which is either a common end– vertex or a crossing. 1 2
The research was supported by the Slovak VEGA grant No. 1/3004/06. Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.049
350
M. Klešˇc / Electronic Notes in Discrete Mathematics 28 (2007) 349–355
The investigation on the crossing number of graphs is a classical and however very difficult problem. The exact value of the crossing number is known only for few specific families of graphs. The Cartesian product is one of few graph classes, for which exact crossing number results are known. Harary at al. [5] conjectured that the crossing number of Cm × Cn is (m − 2)n, for all m, n satisfying 3 ≤ m ≤ n. This has been proved only for m, n satisfying n ≥ m, m ≤ 7. It was recently proved by Glebsky and Salazar [4] that the crossing number of Cm × Cn equals its long–conjectured value at least for n ≥ m(m + 1). Besides of Cartesian product of two cycles, there are several other exact results. In [2,6] the crossing numbers of G × Cn for all graphs G of order four are given. Bokal in [3] confirmed the general conjecture for the crossing number of Cartesian product of path and star formulated in [6]. The table in [7] shows the summary of known crossing numbers for Cartesian products of path, cycle and star with connected graphs of order five. Kulli and Muddebihal [9] gave the characterisation of all pairs of graphs which join is planar graph. It thus seems natural to inquire about crossing numbers of join product of graphs. The join product of two graphs G and H, denoted by G + H, is obtained from vertex–disjoint copies of G and H by adding all edges between V (G) and V (H). For |V (G)| = m and |V (H)| = n, the edge set of G+H is the union of disjoint edge sets of the graphs G, H, and the complete bipartite graph Km,n . It has been long–conjectured in [10] that the crossing number cr(Km,n ) of the complete bipartite graph Km,n equals the Zarankiewicz’s Number Z(m, n) := m−1 m2 n−1 n2 . This conjecture 2 2 has been verified by Kleitman for min{m, n} ≤ 6, see [8]. Using these results, in Section 2, we give the exact values of crossing numbers for join of two paths, join of two cycles, and for join of path and cycle. In Section 3, we give exact values for crossing numbers of G + Pn and G + Cn for all graphs G of order at most four.
2
Paths and cycles
Let Cn and Pn be the cycle and the path with n vertices. Figure 1 shows the graph Cm + Cn . It consists of the graph Km,n and of the edges of two cycles Cm and Cn . There are Z(m, n) = m−1 m2 n−1 n2 crossings 2 2 among the edges of Km,n , and the edges of Cm gross two times the edges of Cn . This implies that cr(Cm + Cn ) ≤ Z(m, n) + 2 . One can easy to see that the deletion of one crossed edge from the cycle Cm in this drawing creates the drawing of the graph Pm + Cn with Z(m, n) + 1 crossings, and the deletion of the crossed edge of Cn in Pm + Cn from the latter drawing
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produces the drawing of the graph Pm + Pn with Z(m, n) crossings. Thus, cr(Pm +Cn ) ≤ Z(m, n)+1 and cr(Pm +Pn ) ≤ Z(m, n) . As, for min{m, n} ≤ 6, cr(Km,n ) = m−1 m2 n−1 n2 , we have the next result. 2 2
Fig. 1. The graph Cm + Cn . Theorem 2.1 cr(Pm + Pn ) = Z(m, n) for any m, n ∈ N with min{m, n} ≤ 6. Let us turn to the graph Pm + Cn . It is obtained from the graph Km,n by adding m − 1 suitable edges of the path Pm and n suitable edges of the cycle Cn . Denote by t1 , t2 , . . . , tm the vertices of Pm in Pm + Cn and let, for i = 1, 2, . . . , m, T i denotes the subgraph of Pm + Cn induced on the edgesincident m i with the vertex ti . We can consider the graph n as Pm ∪Cn ∪( i=1 T ) . mPm +C i Assume now the graph mK1 + Cn = Cn ∪ ( i=1 T ) obtained from the graph Pm + Cn by deleting the edges of Pm . The next Lemma 2.2 enable us to establish crossing numbers for the join products Pm + Cn and Cm + Cn . Lemma 2.2 Let D be a good drawing of mK1 +Cn , m ≥ 2, n ≥ 3, in which no edge of Cn is crossed, and Cn does not separate the other vertices of the graph. Then, for all i, j = 1, 2, . . . , n, two different subgraphs T i and T j cross each other in D at least n2 n−1 times. 2 Proof. By hypothesis, the cycle Cn = c1 c2 . . . cn c1 divides the plane into two regions in such a way that all m vertices of the graph not belonging to Cn lie in one of these regions. Without loss of generality assume that all subgraphs T i , i = 1, 2, . . . , m, are placed in D in the interior region in the view of Cn . For some i ∈ {1, 2, . . . , m}, assume the subgraph Cn ∪ T i of the graph mK1 + Cn and let D be its subdrawing induced from D. As Cn does not cross the edges of T i and no two edges of T i cross each other, the interior region of Cn is divided in D into n triangular regions c1 c2 ti , c2 c3 ti , . . . , cn c1 ti .
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Assume now a subgraph T j , j = i. The vertex tj of the subgraph T j lies in D in one of the triangular regions of D . Without loss of generality assume that tj lies in the triangular region cn c1 ti . As the drawing D is good, the edge {tj , c2 } crosses the edge {ti , c1 } or all edges {ti , cn }, {ti , cn−1 }, . . . , {ti , c3 }. So, the edge {tj , c2 } crosses the edges of T i at least once. The same consideration implies that the edge {tj , cn−1 } crosses the edges of T i at least once, every of the edges {tj , c3 } and {tj , cn−2 } crosses T i at least two times, every of the edges {tj , c4 } and {tj , cn−3 } crosses T i at least three times, . . . . Hence, the total number of crossings between the edges of T i and the edges of T j is at least n n−1 n 2 + 4 + . . . + 2( − 1) = 2 2 2 for even n, and at least n n n n−1 2 + 4 + . . . + 2( − 1) + = 2 2 2 2 if n is odd. This completes the proof.
2
Assume now that there is a drawing of the graph Pm + Cn with less than Z(m, n) + 1 crossings. Since, for min{m, n} ≤ 6, the crossing number of Km,n is Z(m, n), in such a drawing no edge of the cycle Cn is crossed. This implies that, in this drawing, all vertices of Pm are placed in the same region in theview the subdrawing of the cycle Cn . So, by Lemma 2.2, there are at n of n−1 m least 2 2 2 ≥ Z(m, n) + 1 crossings in the drawing. This contradiction with our assumption allows us to formulate the next result. Theorem 2.3 cr(Pm + Cn ) = Z(m, n) + 1 for any m ≥ 2, n ≥ 3 with min{m, n} ≤ 6. Using the similar consideration we have. Theorem 2.4 cr(Cm + Cn ) = Z(m, n) + 2 for any m ≥ 3, n ≥ 3 with min{m, n} ≤ 6.
3
Graphs of small order
For Cartesian product, the crossing numbers of G × Pn and G × Cn for graphs G of small order were studied. One can formulate the same question for join products G + Pn and G + Cn . We studied this question for all graphs G with |V (G)| ≤ 4. In contrast to Cartesian product, where only connected graphs G
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are considered, for join product also disconnected graphs must be taken into account. 3.1
Graphs or order at most 3
There are only three graphs K1 , 2K1 , and K2 = P2 of order less than three. One can easy to verify that, except of the join product P2 + Cn with crossing number 1 (see Theorem 2.3), all other graphs G + Pn and G + Cn are planar. There are four graphs 3K1 , K1 ∪ P2 , P3 , and C3 of order three. In Section 2, the crossing numbers for join products P3 + Pn , P3 + Cn , C3 + Pn , and C3 +Cn are established. For every join product 3K1 +Pn , (K1 ∪P2 )+Pn , 3K1 + Cn , and (K1 ∪ P2 ) + Cn it is possible to find a drawing with at most Z(3, n) crossings and, since every of these graphs contains K3,n as a subgraph, the next Theorem 3.1 holds. Theorem 3.1 For n ≥ 3, cr(3K1 + Pn ) = cr((K1 ∪ P2 ) + Pn ) = cr(3K1 + Cn ) = cr((K1 ∪ P2 ) + Cn ) = Z(3, n). We remark that, for n ≤ 2, the graphs 3K1 + Pn and (K1 ∪ P2 ) + Pn are planar. 3.2
Graphs of order 4
There are eleven graphs Gi of order four drawn in the first column of the Table 1 below. Every join product Gi + Pn and Gi + Cn , i = 1, 2, . . . , 11, contains the graph K4,n as a subgraph and, since cr(K4,n ) = Z(4, n), the crossing number for every graph Gi + Pn and Gi + Cn is at least Z(4, n). For the graphs G1 = 4K1 , G2 = 2K1 ∪ K2 , and G3 = 2K2 there are drawings of the join products Gi + Pn and Gi + Cn with Z(4, n) crossings, and the crossing numbers for all these graphs are established. The same holds for the join product G4 + Pn = (K1 ∪ P3 ) + Pn . The crossing numbers for the join products G5 + Pn = P4 + Pn , G5 + Cn = P4 + Cn , G8 + Pn = C4 + Pn , and G8 + Cn = C4 + Cn were found in Section 2. The proof that the crossing number of the graph G4 + Cn = (K1 ∪ P3 ) + Cn is Z(4, n) + 1 proceeds by using Lemma 2.2 in the same way as the proof of Theorem 2.3. Asano [1] in 1986 proved that the crossing number of the complete tripartite graph K1,3,n is Z(4, n) + n2 . The join product G7 + Pn = K1,3 + Pn contains the graph K1,3,n as a subgraph and, as there is a drawing of the graph K1,3 + Pn with Z(4, n) + n2 crossings, we have cr(K1,3 + Pn ) = Z(4, n) + n2 . Consider now the join product G6 +Pn = (K1 ∪K3 )+Pn . It is not difficult to find a drawing of this graph with Z(4, n) + n2 = 21 n(n − 1) crossings, and
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therefore cr(K1 ∪ K3 ) + Pn ) ≤ 12 n(n − 1). To show that a such drawing is optimal, and that cr(K1 ∪ K3 ) + Pn ) = 12 n(n − 1), needs special proof which proceeds by induction on n with step 2. After this, one can find the crossing number for the join product G9 + Pn . The graph G9 + Pn contains the graph K1 ∪ K3 ) + Pn as a subgraph, and therefore its crossing number is not less than 12 n(n − 1). A drawing of this graph with 12 n(n − 1) crossings confirms that cr(G9 + Pn ) = 12 n(n − 1).
Table 1. Summary of crossing numbers for Gi + Pn and Gi + Cc . We omit proofs which are necessary to show that for the join product G10 +Pn = (K4 −e)+Pn the crossing number is Z(4, n)+ n2 +1 = 12 n(n−1)+1
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and that for the join product G11 + Pn = K4 + Pn the crossing number is Z(4, n) + n2 + n + 1 = 12 n(n + 1) + 1. The known crossing numbers for join products Gi +Pn enable us to establish crossing numbers for the join products Gi +Cn for i = 6, 7, 9, 10, and 11. These proofs we also omit. Table 1 above summarise crossing numbers of the join products Gi + Pn and Gi + Cn for all graphs Gi on four vertices.
References [1] Asano, K., The crossing number of K1,3,n and K2,3,n , J. Graph Theory 10 (1986), 1–8. [2] Beineke, L. W., and R. D. Ringeisen, On the crossing numbers of products of cycles and graphs of order four, J. Graph Theory 4 (1980), 145–155. [3] Bokal, D., On the crossing numbers of Cartesian products with paths, J. Combin. Theory B (to appear). http://lp.fmf.uni-lj.si/~ drago/pathProducts.pdf [4] Glebsky, L. Y., and G. Salazar, The crossing number of Cm ×Cn is as conjectured for n ≥ m(m + 1), J. Graph Treory 47 (2004), 53–72. [5] Harary, F., P. C. Kainen, and A. J. Schwenk, Toroidal graphs with arbitrarily high crossing numbers, Nanta Math. 6 (1973), 58–67. ˇcerbov´a, On the crossing numbers of Sm ×Pn and Sm ×Cn , [6] Jendrol’, S., and M. Sˇ ˇ Casopis pro pˇestov´an´ı matematiky 107 (1982), 225–230. [7] Kleˇsˇc, M., The crossing numbers of Cartesian products of paths with 5–vertex graphs, Discrete Mathematics 233 (2001), 353–359. [8] Kleitman, D. J., The crossing number of K5,n , J. Combin. Theory B 9 (1971), 315–323. [9] Kulli, V. R., and M. H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001), 87-97. [10] Zarankiewicz, K., On a problem of P. Turan concerning graphs, Fund. Math. 41 (1954), 137–145.
The Join of Graphs and Crossing Numbers Mari´an Kleˇsˇc
1,2
Department of Mathematics Faculty of Electrical Engineering and Informatics Technical University 042 00 Koˇsice, Slovak Republic
Abstract It has been long–conjectured by Zarankiewicz that the crossing number of the comm n−1 n plete bipartite graph Km,n equals m−1 2 2 2 2 . This conjecture has been verified by Kleitman for min{m, n} ≤ 6. Using these results, we give the exact values of crossing numbers for join of two paths, join of two cycles, and for join of path and cycle. In addition, we give the exact values of crossing numbers for join products G + Pn and G + Cn for all graphs G of order at most four. Keywords: graph, join product, drawing, crossing number, path, cycle.
1
Introduction
The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane. It is easy to see that a drawing with the minimum number of crossings must be a good drawing; that is, each two edges have at most one point in common, which is either a common end– vertex or a crossing. 1 2
The research was supported by the Slovak VEGA grant No. 1/3004/06. Email: [email protected]
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.01.049
350
M. Klešˇc / Electronic Notes in Discrete Mathematics 28 (2007) 349–355
The investigation on the crossing number of graphs is a classical and however very difficult problem. The exact value of the crossing number is known only for few specific families of graphs. The Cartesian product is one of few graph classes, for which exact crossing number results are known. Harary at al. [5] conjectured that the crossing number of Cm × Cn is (m − 2)n, for all m, n satisfying 3 ≤ m ≤ n. This has been proved only for m, n satisfying n ≥ m, m ≤ 7. It was recently proved by Glebsky and Salazar [4] that the crossing number of Cm × Cn equals its long–conjectured value at least for n ≥ m(m + 1). Besides of Cartesian product of two cycles, there are several other exact results. In [2,6] the crossing numbers of G × Cn for all graphs G of order four are given. Bokal in [3] confirmed the general conjecture for the crossing number of Cartesian product of path and star formulated in [6]. The table in [7] shows the summary of known crossing numbers for Cartesian products of path, cycle and star with connected graphs of order five. Kulli and Muddebihal [9] gave the characterisation of all pairs of graphs which join is planar graph. It thus seems natural to inquire about crossing numbers of join product of graphs. The join product of two graphs G and H, denoted by G + H, is obtained from vertex–disjoint copies of G and H by adding all edges between V (G) and V (H). For |V (G)| = m and |V (H)| = n, the edge set of G+H is the union of disjoint edge sets of the graphs G, H, and the complete bipartite graph Km,n . It has been long–conjectured in [10] that the crossing number cr(Km,n ) of the complete bipartite graph Km,n equals the Zarankiewicz’s Number Z(m, n) := m−1 m2 n−1 n2 . This conjecture 2 2 has been verified by Kleitman for min{m, n} ≤ 6, see [8]. Using these results, in Section 2, we give the exact values of crossing numbers for join of two paths, join of two cycles, and for join of path and cycle. In Section 3, we give exact values for crossing numbers of G + Pn and G + Cn for all graphs G of order at most four.
2
Paths and cycles
Let Cn and Pn be the cycle and the path with n vertices. Figure 1 shows the graph Cm + Cn . It consists of the graph Km,n and of the edges of two cycles Cm and Cn . There are Z(m, n) = m−1 m2 n−1 n2 crossings 2 2 among the edges of Km,n , and the edges of Cm gross two times the edges of Cn . This implies that cr(Cm + Cn ) ≤ Z(m, n) + 2 . One can easy to see that the deletion of one crossed edge from the cycle Cm in this drawing creates the drawing of the graph Pm + Cn with Z(m, n) + 1 crossings, and the deletion of the crossed edge of Cn in Pm + Cn from the latter drawing
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produces the drawing of the graph Pm + Pn with Z(m, n) crossings. Thus, cr(Pm +Cn ) ≤ Z(m, n)+1 and cr(Pm +Pn ) ≤ Z(m, n) . As, for min{m, n} ≤ 6, cr(Km,n ) = m−1 m2 n−1 n2 , we have the next result. 2 2
Fig. 1. The graph Cm + Cn . Theorem 2.1 cr(Pm + Pn ) = Z(m, n) for any m, n ∈ N with min{m, n} ≤ 6. Let us turn to the graph Pm + Cn . It is obtained from the graph Km,n by adding m − 1 suitable edges of the path Pm and n suitable edges of the cycle Cn . Denote by t1 , t2 , . . . , tm the vertices of Pm in Pm + Cn and let, for i = 1, 2, . . . , m, T i denotes the subgraph of Pm + Cn induced on the edgesincident m i with the vertex ti . We can consider the graph n as Pm ∪Cn ∪( i=1 T ) . mPm +C i Assume now the graph mK1 + Cn = Cn ∪ ( i=1 T ) obtained from the graph Pm + Cn by deleting the edges of Pm . The next Lemma 2.2 enable us to establish crossing numbers for the join products Pm + Cn and Cm + Cn . Lemma 2.2 Let D be a good drawing of mK1 +Cn , m ≥ 2, n ≥ 3, in which no edge of Cn is crossed, and Cn does not separate the other vertices of the graph. Then, for all i, j = 1, 2, . . . , n, two different subgraphs T i and T j cross each other in D at least n2 n−1 times. 2 Proof. By hypothesis, the cycle Cn = c1 c2 . . . cn c1 divides the plane into two regions in such a way that all m vertices of the graph not belonging to Cn lie in one of these regions. Without loss of generality assume that all subgraphs T i , i = 1, 2, . . . , m, are placed in D in the interior region in the view of Cn . For some i ∈ {1, 2, . . . , m}, assume the subgraph Cn ∪ T i of the graph mK1 + Cn and let D be its subdrawing induced from D. As Cn does not cross the edges of T i and no two edges of T i cross each other, the interior region of Cn is divided in D into n triangular regions c1 c2 ti , c2 c3 ti , . . . , cn c1 ti .
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Assume now a subgraph T j , j = i. The vertex tj of the subgraph T j lies in D in one of the triangular regions of D . Without loss of generality assume that tj lies in the triangular region cn c1 ti . As the drawing D is good, the edge {tj , c2 } crosses the edge {ti , c1 } or all edges {ti , cn }, {ti , cn−1 }, . . . , {ti , c3 }. So, the edge {tj , c2 } crosses the edges of T i at least once. The same consideration implies that the edge {tj , cn−1 } crosses the edges of T i at least once, every of the edges {tj , c3 } and {tj , cn−2 } crosses T i at least two times, every of the edges {tj , c4 } and {tj , cn−3 } crosses T i at least three times, . . . . Hence, the total number of crossings between the edges of T i and the edges of T j is at least n n−1 n 2 + 4 + . . . + 2( − 1) = 2 2 2 for even n, and at least n n n n−1 2 + 4 + . . . + 2( − 1) + = 2 2 2 2 if n is odd. This completes the proof.
2
Assume now that there is a drawing of the graph Pm + Cn with less than Z(m, n) + 1 crossings. Since, for min{m, n} ≤ 6, the crossing number of Km,n is Z(m, n), in such a drawing no edge of the cycle Cn is crossed. This implies that, in this drawing, all vertices of Pm are placed in the same region in theview the subdrawing of the cycle Cn . So, by Lemma 2.2, there are at n of n−1 m least 2 2 2 ≥ Z(m, n) + 1 crossings in the drawing. This contradiction with our assumption allows us to formulate the next result. Theorem 2.3 cr(Pm + Cn ) = Z(m, n) + 1 for any m ≥ 2, n ≥ 3 with min{m, n} ≤ 6. Using the similar consideration we have. Theorem 2.4 cr(Cm + Cn ) = Z(m, n) + 2 for any m ≥ 3, n ≥ 3 with min{m, n} ≤ 6.
3
Graphs of small order
For Cartesian product, the crossing numbers of G × Pn and G × Cn for graphs G of small order were studied. One can formulate the same question for join products G + Pn and G + Cn . We studied this question for all graphs G with |V (G)| ≤ 4. In contrast to Cartesian product, where only connected graphs G
M. Klešˇc / Electronic Notes in Discrete Mathematics 28 (2007) 349–355
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are considered, for join product also disconnected graphs must be taken into account. 3.1
Graphs or order at most 3
There are only three graphs K1 , 2K1 , and K2 = P2 of order less than three. One can easy to verify that, except of the join product P2 + Cn with crossing number 1 (see Theorem 2.3), all other graphs G + Pn and G + Cn are planar. There are four graphs 3K1 , K1 ∪ P2 , P3 , and C3 of order three. In Section 2, the crossing numbers for join products P3 + Pn , P3 + Cn , C3 + Pn , and C3 +Cn are established. For every join product 3K1 +Pn , (K1 ∪P2 )+Pn , 3K1 + Cn , and (K1 ∪ P2 ) + Cn it is possible to find a drawing with at most Z(3, n) crossings and, since every of these graphs contains K3,n as a subgraph, the next Theorem 3.1 holds. Theorem 3.1 For n ≥ 3, cr(3K1 + Pn ) = cr((K1 ∪ P2 ) + Pn ) = cr(3K1 + Cn ) = cr((K1 ∪ P2 ) + Cn ) = Z(3, n). We remark that, for n ≤ 2, the graphs 3K1 + Pn and (K1 ∪ P2 ) + Pn are planar. 3.2
Graphs of order 4
There are eleven graphs Gi of order four drawn in the first column of the Table 1 below. Every join product Gi + Pn and Gi + Cn , i = 1, 2, . . . , 11, contains the graph K4,n as a subgraph and, since cr(K4,n ) = Z(4, n), the crossing number for every graph Gi + Pn and Gi + Cn is at least Z(4, n). For the graphs G1 = 4K1 , G2 = 2K1 ∪ K2 , and G3 = 2K2 there are drawings of the join products Gi + Pn and Gi + Cn with Z(4, n) crossings, and the crossing numbers for all these graphs are established. The same holds for the join product G4 + Pn = (K1 ∪ P3 ) + Pn . The crossing numbers for the join products G5 + Pn = P4 + Pn , G5 + Cn = P4 + Cn , G8 + Pn = C4 + Pn , and G8 + Cn = C4 + Cn were found in Section 2. The proof that the crossing number of the graph G4 + Cn = (K1 ∪ P3 ) + Cn is Z(4, n) + 1 proceeds by using Lemma 2.2 in the same way as the proof of Theorem 2.3. Asano [1] in 1986 proved that the crossing number of the complete tripartite graph K1,3,n is Z(4, n) + n2 . The join product G7 + Pn = K1,3 + Pn contains the graph K1,3,n as a subgraph and, as there is a drawing of the graph K1,3 + Pn with Z(4, n) + n2 crossings, we have cr(K1,3 + Pn ) = Z(4, n) + n2 . Consider now the join product G6 +Pn = (K1 ∪K3 )+Pn . It is not difficult to find a drawing of this graph with Z(4, n) + n2 = 21 n(n − 1) crossings, and
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therefore cr(K1 ∪ K3 ) + Pn ) ≤ 12 n(n − 1). To show that a such drawing is optimal, and that cr(K1 ∪ K3 ) + Pn ) = 12 n(n − 1), needs special proof which proceeds by induction on n with step 2. After this, one can find the crossing number for the join product G9 + Pn . The graph G9 + Pn contains the graph K1 ∪ K3 ) + Pn as a subgraph, and therefore its crossing number is not less than 12 n(n − 1). A drawing of this graph with 12 n(n − 1) crossings confirms that cr(G9 + Pn ) = 12 n(n − 1).
Table 1. Summary of crossing numbers for Gi + Pn and Gi + Cc . We omit proofs which are necessary to show that for the join product G10 +Pn = (K4 −e)+Pn the crossing number is Z(4, n)+ n2 +1 = 12 n(n−1)+1
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355
and that for the join product G11 + Pn = K4 + Pn the crossing number is Z(4, n) + n2 + n + 1 = 12 n(n + 1) + 1. The known crossing numbers for join products Gi +Pn enable us to establish crossing numbers for the join products Gi +Cn for i = 6, 7, 9, 10, and 11. These proofs we also omit. Table 1 above summarise crossing numbers of the join products Gi + Pn and Gi + Cn for all graphs Gi on four vertices.
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