The crossing number of the Cartesian product of paths with complete graphs

The crossing number of the Cartesian product of paths with complete graphs

Discrete Mathematics 328 (2014) 71–78 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc ...

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Discrete Mathematics 328 (2014) 71–78

Contents lists available at ScienceDirect

Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

The crossing number of the Cartesian product of paths with complete graphs ZhangDong Ouyang a,∗ , Jing Wang b , YuanQiu Huang c a

Department of Mathematics, Hunan First Normal University, 410205 Changsha, China

b

Department of Mathematics and Information Sciences, Changsha University, 410003 Changsha, China

c

Department of Mathematics, Hunan Normal University, 410081 Changsha, China

article

info

Article history: Received 20 December 2012 Received in revised form 4 March 2014 Accepted 17 March 2014 Available online 18 April 2014 Keywords: Crossing number Cartesian product Zip product Complete graph

abstract In this paper, we determine the crossing number of Km \ e by the construction method for m ≤ 12 and apply the zip product to obtain that cr (Km Pn ) = (n − 1)cr (Km+2 \ e) + 2cr (Km+1 ) for n ≥ 1. Furthermore, we have

cr (Km Pn ) =



1 m+1 4

2



m−1 2



m−2 2

  n

m+4 2

   m−4 + 2

for n ≥ 1, 1 ≤ m ≤ 10, which is consistent with Zheng’s conjecture for the crossing number of Km Pn . © 2014 Elsevier B.V. All rights reserved.

1. Introduction All graphs considered here are simple, finite and undirected and are also connected. For graph theory terminology not defined here, we direct the reader to [7]. A drawing of a graph G = (V , E ) is a mapping φ that assigns to each vertex in V a distinct point in the plane and to each edge uv in E a continuous arc (i.e., a homeomorphic image of a closed interval) connecting φ(u) and φ(v), without passing through the image of any other vertex. In addition, we impose the following conditions on a drawing: (1) no three edges have an interior point in common, (2) if two edges share an interior point p, then they cross at p, and (3) any two edges of a drawing have only a finite number of crossings (common interior points). The crossing number cr (G) of a graph G is the minimum number of edge crossings in any drawing of G. Let D be a drawing of the graph G, and we denote the number of crossings in D by crD (G). For more on the theory of crossing numbers, we refer the reader to [8]. The Cartesian product GH of graphs G and H has the vertex set V (G) × V (H ) and the edge set E (GH ) = {(x1 , y1 )(x2 , y2 )|x1 = x2 and y1 y2 ∈ E (H ) or y1 = y2 and x1 x2 ∈ E (G)}. The investigation of the crossing number of a graph is a classical but very difficult problem (for example, see [8]). In fact, computing the crossing number of a graph is NP-complete [9], and the exact values are known only for very restricted classes of graphs. The crossing numbers of the Cartesian products of graphs have been studied since 1973, when Harary et al. [12] conjectured that cr (Cm Cn ) = (m − 2)n for 3 ≤ m ≤ n. This conjecture has been verified in [1,20–23] for m ≤ 7, n ≥ m. Glebsky and Salazar [10] also showed that the conjecture holds for n ≥ m(m + 1) and m ≥ 3. Klešč [15] determined the crossing numbers of the products of all 4-vertex graphs with paths and stars except cr (K1,3 Pn ), which was earlier determined by Jendroľ and Ščerbová [13], who also obtained cr (K1,3 Cn ) for n ≥ 1. In their paper, they



Corresponding author. E-mail address: [email protected] (Z. Ouyang).

http://dx.doi.org/10.1016/j.disc.2014.03.011 0012-365X/© 2014 Elsevier B.V. All rights reserved.

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1 conjectured that cr (Sm Pn ) = (n − 1)⌊ m ⌋⌊ m− ⌋ for m, n ≥ 1. For general n, the conjecture was recently confirmed by 2 2 Bokal in [5]. Beineke and Ringeisen [3,4] determined the crossing numbers of the products of all 4-vertex graphs with cycles. Klešč [16–18] determined the crossing numbers of the products of all 5-vertex graphs with paths. In particular, he proved that cr (K5 Pn ) = 6n for n ≥ 1 in [16]. Zheng et al. [26] recently proved that cr (K6 Pn ) = 15n + 3 for n ≥ 1 and

cr (Km Pn ) ≥ (n − 1)cr (Km+2 \ e) + 2cr (Km+1 ).

(1.1)

In their paper, for n ≥ 1, they conjecture that cr (Km Pn ) =



1 m+1 4



m−1

2



m−2

2

  n

2

   m−4 + .

m+4 2

2

(1.2)

In this contribution, we show that equality holds in (1.1) for m ≥ 1 and conjecture that (1.2) holds for 1 ≤ m ≤ 10. The approach is seemingly new. To obtain the crossing number of Km \ e, we construct a drawing of Km from the drawing of Km \ e and obtain two lower bound expressions of cr (Km \ e) by the standard counting method used in [14,25]. To prove that equality holds in (1.1), we introduce the zip product operation that was used in [2,5,6,24] and prove a lemma about it with similar sufficient conditions to the ones in [5]. 2. Some definitions and lemmas Definition 2.1. For a graph G, let A, B ⊆ E (G); then, for a drawing φ of G, let crφ (A, B) =



|φ(a) ∩ φ(b)|.

a∈A,b∈B

Additionally, let crφ (A, A) = crφ (A). Informally, crφ (A, B) denotes the number of crossings between every pair of edges where one edge is in A and the other in B. For three mutually disjoint subsets A, B, C ⊂ E (G), the identities crφ (A ∪ B) = crφ (A) + crφ (B) + crφ (A, B)

(2.1)

crφ (A, B ∪ C ) = crφ (A, B) + crφ (B, C )

(2.2)

and

are noted. Let Gi , i = 1, 2, be a graph with a vertex vi ∈ V (Gi ) whose neighborhood Ni = NGi (vi ) has size d. A zip function of graphs G1 and G2 at vertices v1 and v2 is a bijection σ : N1 → N2 . The zip product G1 ⊙σ G2 of graphs G1 and G2 according to σ is obtained from the disjoint union of G1 − v1 and G2 − v2 by adding the edges uσ (u), u ∈ N1 . A drawing Di of the graph Gi (i = 1, 2) defines (up to a circular permutation) a bijection πi : Ni → {1, 2, . . . , d} that respects the edge rotation around vi in Di . The zip function of drawings D1 and D2 at vertices v1 and v2 is σ : N1 → N2 , σ = π2−1 π1 . The zip product D1 ⊙σ D2 of D1 and D2 according to σ is obtained from D1 by adding a mirrored copy of D2 that has v2 incident with the infinite face disjointly into some face of D1 incident with v1 by removing vertices v1 and v2 and small disks around them from the drawings and then joining the edges according to the function σ . For more detail, we refer the reader to [5,6]. For this construction, the following lemmas hold: Lemma 2.1 ([5]). For i = 1, 2, let Di be an optimal drawing of Gi , let vi ∈ V (Gi ) be a vertex of degree d, and let σ be a zip function of D1 and D2 at v1 and v2 . Then, cr (G1 ⊙σ G2 ) ≤ cr (G1 ) + cr (G2 ). Let Sn = K1,n be a star graph with n vertices of degree 1 (called the leaves of the star) and one vertex of degree n (the center). Let G be a graph and S ⊆ V (G), |S | = n. We say that S is k-star-connected in G if there exist k disjoint sets F1 , F2 , . . . , Fk ⊆ E (G) such that either G[Fi ] is a subdivision of Sn with S being the leaves or G[Fi ] is a subdivision of Sn−1 with all its leaves and the center belonging to S. Lemma 2.2 ([5]). Let G1 and G2 be disjoint graphs, vi ∈ V (Gi ), deg (vi ) = d, and let the neighborhood Ni of vi be 2-star-connected in Gi − vi , i = 1, 2. Then, cr (G1 ⊙σ G2 ) ≥ cr (G1 ) + cr (G2 ) for any bijection σ : N1 → N2 . Lemma 2.3 ([26]). cr (Km \ e) ≤

1 4



m+2 2



m−1 2



m−3 2



m−4 2

 .

Some of the proofs in this paper are based on these results for the crossing numbers of complete graphs, more precisely as follows: Conjecture 2.1 ([11]). cr (Km ) =

1 4

  m 2

m−1 2



m−2 2



m−3 2



.

It has been proven by Guy [11] for m ≤ 10 and by Pan and Richter [19] for m = 11, 12, respectively.

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73

Fig. 1. The local situation of D.

3. Lower bounds for cr (Km \ e) First, we give the following lower bound of the crossing number of Km \ e in terms of the crossing numbers of Km−1 and Km . Theorem 3.1. cr (Km \ e) ≥

1 m



   m−3 2cr (Km−1 ) + (m − 2) cr (Km ) − , for m ≥ 2. 2

Proof. The graph Km \ e has m − 2 vertices of degree m − 1, denoted by vi , i = 1, 2, . . . , m − 2, and two vertices of degree m − 2, denoted by x and y, respectively. Let T x and T y denote the subgraphs induced by m − 2 edges incident with the vertex x and y, respectively. It is easy to see that the subgraph induced by m − 2 vertices v1 , v2 , . . . , vm−2 is isomorphic to the complete graph Km−2 . Thus, we have Km \ e = Km−2 ∪ T x ∪ T y . y y Let Kmx −1 = Km−2 ∪ T x and Km−1 = Km−2 ∪ T y , respectively. It is easy to see that Kmx −1 ∼ = Km−1 and Km−1 ∼ = Km−1 . Let D be an optimal drawing of Km \ e. Using (2.1) and (2.2), we have

crD (Km \ e) = crD (Kmx −1 ) + crD (T y , Km−2 ) + crD (T x , T y ); crD (Km \ e) =

y crD Km−1

(

) + crD (T , Km−2 ) + crD (T , T ). x

x

y

(3.1) (3.2)

Summing (3.1) and (3.2), we obtain y

2crD (Km \ e) = crD (Kmx −1 ) + crD (Km−1 ) + crD (T x ∪ T y , Km−2 ) + 2crD (T x , T y ).

(3.3)

For i = 1, 2, . . . , m − 2, ai denote the numbers of crossings on the path xvi y. We can see that in the plane R2 , there exists a circle neighborhood N (D(vi ), ε) = {s ∈ R2 : ∥s − D(vi )∥ ≤ ε}, where ε is a sufficiently small positive number such that for any other edge e′ of Km \ e incident with vi , no crossing appears on the segment D(e′ ) ∩ N (D(vi ), ε). For 1 ≤ i ≤ m − 2, we will obtain a drawing Di of Km from D. For this purpose, we draw a new edge from x to y (written as xiy): first depart from vertex x near the edge xvi , then bypass vertex vi in N (D(vi ), ε), and finally connect to y near to the edge yvi (see Fig. 1, where the circuit C denotes the boundary of N (D(vi ), ε)). It is not difficult to see that the edge xiy can cross the edge of Km \ e exactly ai + mi times in Di , where mi denotes the minimal number of edges that lie on the same side 3 ⌋. Thus, of path xvi y. Clearly, mi ≤ ⌊ m− 2 ai = crDi (Km ) − crD (Km \ e) − mi .

(3.4)

By implication, m−2

 i=1

m−2

ai =



m−2

crDi (Km ) − (m − 2)crD (Km \ e) −

i =1



mi .

(3.5)

i=1

Additionally, it is easy to see that m−2



ai = crD (Km−2 , T x ∪ T y ) + 2crD (T x , T y ).

(3.6)

i=1

By combining (3.3) and (3.6), we obtain m−2 y

2crD (Km \ e) = crD (Kmx −1 ) + crD (Km−1 ) +

 i=1

ai .

(3.7)

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Again, from (3.5) and (3.7), we have m−2 y

mcrD (Km \ e) = crD (Kmx −1 ) + crD (Km−1 ) +



m−2

crDi (Km ) −



i =1

≥ 2cr (Km−1 ) + (m − 2)cr (Km ) − (m − 2) as stated.

mi

i =1



m−3

 (3.8)

2



Theorem 3.2. cr (Km \ e) ≥

m+2 cr m−2

(Km−1 ), for m ≥ 3.

Proof. We will use the same notation as in Theorem 3.1. From (3.8), we take that m−2 y

mcrD (Km \ e) = crD (Kmx −1 ) + crD (Km−1 ) +



m−2

crDi (Km ) −

i=1



mi .

(3.9)

i=1

y

Set crD (Kmx −1 ) = cr (Km−1 ) + l1 , crD (Km−1 ) = cr (Km−1 ) + l2 , where l1 ≥ 0, l2 ≥ 0, crDi (Km ) = cr (Km ) + ci , and ci ≥ 0, 1 ≤ i ≤ m − 2. Thus, equality (3.9) can be written as m−2

mcrD (Km \ e) = 2cr (Km−1 ) + l1 + l2 + (m − 2)cr (Km ) +



m−2

ci −

i =1



mi .

(3.10)

i=1

Define the responsibility, rφ (v), of a vertex v in a drawing φ as the total number of crossings on all edges  incident to that vertex. Because each crossing is in the responsibility of 4 vertices, the total responsibility of all vertices rφ (v) = 4cr (φ). For more detail, we refer the reader to [11]. Thus, it is clear that y

rD (x) = crD (Km \ e) − crD (Km−1 )

= crD (Km \ e) − cr (Km−1 ) − l2 ,

(3.11)

and rD (y) = crD (Km \ e) − crD (Kmx −1 )

= crD (Km \ e) − cr (Km−1 ) − l1 .

(3.12)

Similarly, for 1 ≤ i ≤ m − 2, rDi (vi ) = crDi (Km ) − crDi (Km \ vi )

≤ cr (Km ) + ci − cr (Km−1 ).

(3.13)

However, note also that rDi (vi ) = rD (vi ) + mi .

(3.14)

Using (3.13) and (3.14), we obtain rD (vi ) ≤ cr (Km ) − cr (Km−1 ) + ci − mi .

(3.15)

Therefore, it now follows from the definition of responsibility, together with (3.11), (3.12) and (3.15), that m−2

4crD (Km \ e) = rD (x) + rD (y) +



rD (vi )

i =1 m−2

≤ 2crD (Km \ e) − 2cr (Km−1 ) − (l1 + l2 ) + (m − 2)cr (Km ) − (m − 2)cr (Km−1 ) +

 i =1

m−2

ci −



mi ,

i =1

which yields m−2

2crD (Km \ e) ≤ (m − 2)cr (Km ) − mcr (Km−1 ) − (l1 + l2 ) +

 i =1

Therefore, by combining (3.10) and (3.16), we have

(m − 2)crD (Km \ e) ≥ (m + 2)cr (Km−1 ) + 2(l1 + l2 );

m−2

ci −

 i=1

mi .

(3.16)

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75

Table 1 The known crossing numbers for Km and Km \ e. m

≤4

5

6

7

8

9

10

11

12

cr (Km ) cr (Km \ e)

0 0

1 0

3 2

9 6

18 15

36 30

60 54

100 90

150 140

that is to say, we obtain the following inequality m+2

crD (Km \ e) ≥

m−2 m+2

≥ as desired.

m−2

cr (Km−1 ) +

2(l1 + l2 ) m−2

cr (Km−1 )



Theorem 3.3. If Conjecture 2.1 is true for m = 2M − 1, where M ≥ 2, then the equality holds in Theorem 3.2 for m = 2M, i.e., cr (K2M \ e) =

1 4

(M 2 − 1)(M − 2)2 .

Proof. Lemma 2.3 clearly suffices to prove cr (K2M \ e) ≥

1 4

(M 2 − 1)(M − 2)2 .

(3.17)

If Conjecture 2.1 is true for m = 2M − 1, then cr (K2M −1 ) =

=



1 2M − 1 4 1 4



2M − 2

2



2M − 3

2

2



2M − 4



2

(M − 1)2 (M − 2)2 .

Therefore, inequality (3.17) follows from Theorem 3.2, and the proof is completed. Theorem 3.4. cr (Km \ e) =

1 4



m+2 2



m−1 2



m−3 2





 m−4 , for 1 ≤ m ≤ 12. 2

Proof. The cases for m ≤ 5 are trivial. Because Conjecture 2.1 is true for m ≤ 12, the theorem follows from Theorem 3.3 for m = 6, 8, 10, 12. Assume now m = 7. Then, Lemma 2.3 shows that cr (K7 \ e) ≤ 6. Moreover, applying Theorem 3.1, we have cr (K7 \ e) ≥

=

1

(2cr (K6 ) + 5(cr (K7 ) − 2)) 7 41 7

.

Additionally, note that cr (K7 \ e) is an integer, which implies that cr (K7 \ e) ≥ 6, and the theorem holds for m = 7. The proofs for m = 9, 11 are similar to the proof for m = 7, and the details are left to the reader.  Remark 3.1. As an intuitive aid, in Table 1, we summarize the known crossing numbers for Km and Km \ e. 4. Zip product Let G be a graph and S ⊆ V (G), |S | = n. We say that S is k-strict-star-connected in G if there exist k different sets F1 , F2 , . . . , Fk ⊆ E (G) (possibly with common edges) such that either G[Fi ] is a star Sn with S as the leaves or G[Fi ] is a star Sn−1 with all its leaves and the center belonging to S. We call each Fi a star-set of S. Lemma 4.1. Let G1 and G2 be disjoint graphs, vi ∈ V (Gi ), deg (vi ) = d, and let the neighborhood Ni of vi be 4-strict-starconnected in Gi − vi , i = 1, 2. Then, cr (G1 ⊙σ G2 ) ≥ cr (G1 ) + cr (G2 ) for any bijection σ : N1 → N2 . Proof. Let F = {uσ (u)|u ∈ NG1 (v1 )} be the set of edges joining G1 − v1 and G2 − v2 in G = G1 ⊙σ G2 . For i = 1, 2, j = 1, 2, 3, 4, let Fij ⊆ Ei = E (Gi − vi ) be different star-sets of Ni . Let G1j be the subgraph of G generated with the edges of E1 , F2j and F . We define the graph G2j similarly. Clearly, Gij is isomorphic to a subdivision of Gi , which implies that cr (Gij ) = cr (Gi ).

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Z. Ouyang et al. / Discrete Mathematics 328 (2014) 71–78

Fig. 2. G01 and G0 .

Let D be an optimal drawing of G. It is not difficult to show that cr (G1 ) ≤ crD (E1 , E1 ) + crD (E1 , F ) + crD (E1 , F2j ).

(4.1)

By summing (4.1) for 1 ≤ j ≤ 4 and noting that any e ∈ E2 is contained in at most two F2j ⊆ E2 , we have 4cr (G1 ) ≤ 4crD (E1 , E1 ) + 4crD (E1 , F ) +

4 

crD (E1 , F2j )

j =1

 ≤ 4crD (E1 , E1 ) + 4crD (E1 , F ) + 2crD E1 ,

4 

 .

(4.2)

.

(4.3)

F2j

j =1

Similarly, we have

 4cr (G2 ) ≤ 4crD (E2 , E2 ) + 4crD (E2 , F ) + 2crD

E2 ,

4 

 F1j

j =1

Combining inequalities (4.2) and (4.3), we have 4cr (G1 ) + 4cr (G2 ) ≤ 4crD (E1 , E1 ) + 4crD (E2 , E2 ) + 4crD (E1 ∪ E2 , F )

 + 2crD E1 ,

4  j =1

 F2j

 + 2crD E2 ,

4 

 F1j

j =1

≤ 4crD (E1 , E1 ) + 4crD (E2 , E2 ) + 4crD (E1 ∪ E2 , F ) + 2crD (E1 , E2 ) + 2crD (E2 , E1 ) = 4crD (G) − 4crD (F ) ≤ 4cr (G). Thus, the proof is completed.



Remark 4.1. We can find that star-connected is different from strict-star-connected according to their definitions. Starconnected star-sets are pairwise disjoint, which is not required for strict-star-connected. However, it seems that requiring 4-strict-star-connected in Lemma 4.1 is too strong a condition. We propose the following question: Question 4.1. Does k-strict-star-connected (k ≤ 3) imply the claim of Lemma 4.1 ? 5. The crossing number of Km Pn By G + nK1 , we denote a joint product of G, that is, the graph obtained from G by adding n vertices vi and the edges vvi for each v ∈ V (G), i = 0, 1, . . . , n − 1. For simpler labeling, let G0 and G01 denote the graphs G + K1 and G + 2K1 , respectively (see Fig. 2). We call a vertex v ∈ V (G) a dominating vertex of G if it is adjacent to all other vertices in G. By Pn , we denote ˆ Pn , we denote the capped Cartesian product of G and Pn , i.e., the the path of length n whose vertices are 0, 1, . . . , n. With G graph obtained from GPn by adding two new vertices v0′ and vn′ and connecting v0′ with all the vertices of G{0} and vn′ with all the vertices of G{n} in GPn .

ˆ Pn ) = (n + 1)cr (G + 2K1 ) for n ≥ 0. Lemma 5.1 ([5]). Let G be a graph with a dominating vertex. Then, cr (G ˆ Pn ) ≤ (n + 1)cr (G + 2K1 ) for n ≥ 0. Lemma 5.2. Let G be a graph. Then, cr (G

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77

Proof. Let D be an optimal drawing of G01 . For i = 0, 1, let πi be the circular labeling of the vertices of V (G) around vi in D. The vertex vi has πi−1 as the circular labeling of its neighborhood in the mirror drawing D′ . Set D0 = D. For odd n, let Dn = Dn−1 ⊙σ D′ using a vertex with labeling π1 and a vertex with labeling π1−1 , and for even n, let Dn = Dn−1 ⊙σ D using a ˆ Pn . However, the definition of a vertex with labeling π0−1 and a vertex with labeling π0 . It is clear that Dn is a drawing of G zip function implies that cr (Dn ) = (n + 1)cr (D) = (n + 1)cr (G + 2K1 ). The lemma follows.  Theorem 5.1. Let G be a vertex transitive graph. Then, for n ≥ 1, cr (GPn ) ≤ (n − 1)cr (G + 2K1 ) + 2cr (G + K1 ).

ˆ Pk and G0 , respectively. We will obtain a drawing Dn of GPn from Dn−2 Proof. Let Dk and D be an optimal drawing of G and D. For n = 1, set D1 = D ⊙σ 0 D, where ⊙σ 0 is a zip function of D and D according to vertices v0 and v0 . The definition of 0

0

a zip function implies that cr (D1 ) = 2cr (D) = 2cr (G + K1 ). However, it is clear that D1 is a drawing of GP1 . Thus, the theorem follows for n = 1. For n ≥ 2, set Dn = (D ⊙σ 0 Dn−2 ) ⊙σ 0 D, where σ00 is a zip function of D and Dn−2 according to v0 and v0′ and σn0−2 is n−2

0

a zip function of D ⊙σ 0 Dn−2 and D according to vn′ −2 and v0 . As G is a vertex transitive graph, it is not difficult to show that 0 Dn is a drawing of GPn . It follows from the definition of a zip function and Lemma 5.2 that cr (GPn ) ≤ cr (Dn )

= cr (D) + cr (Dn−2 ) + cr (D) ≤ 2cr (G + K1 ) + (n − 1)cr (G + 2K1 ) as desired.



Theorem 5.2. Let G be a graph with four dominating vertices. Then, for n ≥ 1, cr (GPn ) ≥ (n − 1)cr (G + 2K1 ) + 2cr (G + K1 ). Proof. For n = 1, it is not difficult to show that GP1 = G0 ⊙σ 0 G0 , where σ00 : N0 → N0 maps a vertex v ∈ N0 = NG0 (v0 ) to 0

its counterpart in N0 . Note that v0 ∈ V (G0 ) have 4-strict-star-connected neighborhoods in G0 , and Lemma 4.1 implies that cr (GP1 ) ≥ 2cr (G0 ) = 2cr (G + K1 ).

ˆ Pn−2 )) ⊙σ 0 For n ≥ 2, we have GPn = (G0 ⊙σ 0 (G

n−2

0

ˆ

G0 , where σ00 : N0 → Nˆ 0 maps a vertex v ∈ N0 to its counterpart in

ˆ

Nˆ 0 = NGˆ Pn−2 (v0′ ), and σn0−2 : Nˆ n−2 → N0 maps a vertex v ∈ Nˆ n−2 = NG0 ⊙ 0 (Gˆ Pn−2 ) (vn′ −2 ) to its counterpart in N0 . The proof σ 0

of this observation is merely technical and is left to the reader. As G has four dominating vertices, the vertices v0 ∈ V (G0 ) ˆ Pk ). Thus, it follows from Lemmas 4.1 have 4-strict-star-connected neighborhoods in G0 , the same holds for v0′ , vk′ ∈ V (G and 5.1 that

ˆ Pn−2 )) ⊙σ 0 cr (GPn ) = cr ((G0 ⊙σ 0 (G

n−2

0

G0 )

≥ 2cr (G0 ) + cr (Gˆ Pn−2 ) = 2cr (G + K1 ) + (n − 1)cr (G + 2K1 ) as stated.



Corollary 5.1. For m ≥ 4, n ≥ 1, we have cr (Km Pn ) = (n − 1)cr (Km+2 \ e) + 2cr (Km+1 ). Proof. It is clear that Km + 2K1 ∼ = Km+2 \ e and Km + K1 ∼ = Km+1 . This claim is an immediate consequence of Theorems 5.1 and 5.2.  Corollary 5.2. For n ≥ 1 and 1 ≤ m ≤ 10, we have cr (Km Pn ) =



1 m+1 4

2



m−1 2



m−2 2

  n

m+4 2

   m−4 + . 2

Proof. The cases for m ≤ 3 are trivial. Because Conjecture 2.1 is true for m ≤ 12, the claim follows immediately from Theorem 3.4 and Corollary 5.1 for m ≥ 4. 

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Corollary 5.3. If Conjecture 2.1 is true for m = 2M + 1, where M ≥ 6, then, for n ≥ 1, cr (K2M Pn ) =

1 4

M (M − 1)2 (nM + M + 2n − 2).

Proof. This claim is an immediate consequence of Theorem 3.3 and Corollary 5.1.



Acknowledgments The authors are indebted to two anonymous referees for their Suggestions, which improved the presentation and made it more readable. The work was supported by the National Natural Science Foundation of China (No. 11301169 & 11371133), Scientific Research Fund of Hunan Provincial Education Department (No. 12B026) and Hunan Provincial Natural Science Foundation of China (No. 13JJ4110). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

J. Adamsson, R.B. Richter, Arrangements, circular arrangements and the crossing number of C7 Cn , J. Combin. Theory Ser. B 90 (2004) 21–39. L. Beaudou, D. Bokal, On the sharpness of some results relating cuts and crossing numbers, Electron. J. Combin. 17 (2010) #R96. L.W. Beineke, R.D. Ringeisen, On the crossing numbers of products of cycles and graphs of order four, J. Graph Theory 4 (1980) 145–155. L.W. Beineke, R. Wilson, Selected Topics in Graph Theory, Academic, New York, 1978, pp. 68–72. D. Bokal, On the crossing number of Cartesian procuct with path, J. Combin. Theory Series B 97 (2007) 381–384. D. Bokal, On the crossing numbers of Cartesian products with trees, J. Graph Theory 56 (4) (2007) 287–300. J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan Press Ltd, London, 1976. P. Erdös, R.K. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) 52–58. M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebr. Discrete Methods 4 (1983) 312–316. L.Y. Glebsky, G. Salazar, The crossing number of Cm Cn is as conjectured for n ≥ m(m + 1), J. Graph Theory 47 (2004) 53–72. R.K. Guy, Crossing numbers of graphs, in: Graph Theory and Applications, in: Lecture Notes in Mathematics, vol. 303, Springer-Verlag, Heidelberg, Berlin, New York, 1972, pp. 111–124. [12] F. Harary, P.C. Kainen, A.J. Schwenk, Toroidal graphs with arbitrarily high crossing numbers, Nanta Math. 6 (1973) 58–67.

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

S. Jendroľ, M. Ščerbová, On the crossing numbers of Sm Pn and Sm Cn , Čas. Pest. Mat. 107 (1982) 225–230. D.J. Kleitman, The crossing number of K5,n , J. Combin. Theory 9 (1971) 315–323. M. Klešč, The crossing numbers of products of paths and stars with 4-vertex graphs, J. Graph Theory 6 (1994) 605–614. M. Klešč, The crossing numbers of K5 Pn , Tatra Mt. Math. Publ. 18 (1999) 63–68. M. Klešč, The crossing numbers of Cartesian products of paths with 5-vertex graphs, Discrete Math. 233 (2001) 353–359. M. Klešč, On the crossing numbers of products of stars and graphs of order five, Graphs Combin. 17 (2001) 289–294. S.J. Pan, R.B. Richter, The crossing number of K11 is 100, J. Graph Theory 56 (2007) 128–134. R.B. Richter, C. Thomassen, Intersections of curve system and the crossing number of C5 C5 , Discrete Comput. Geom. 13 (1995) 149–153. R.D. Ringeisen, L.W. Beineke, The crossing number of C3 Cn , J. Combin. Theory 24 (1978) 134–136. G. Salazar, Drawings of Cm Cn with one disjoint family, J. Combin. Theory Ser. B 76 (1999) 21–39. F. Shahrokhi, O. Sýkora, L.A. Székely, I. Vrt’o, Intersection of curves and crossing number of Cm Cn on surfaces, Discrete Comput. Geom. 19 (1998) 237–247. [24] L. Tang, S.X. Lv, Y.Q. Huang, The Crossing number of Cartesian products of complete bipartite graphs K2,m with paths Pn , Graphs Combin. 23 (6) (2007) 659–666. [25] D.R. Woodall, Cyclic-order graphs and Zarankiewicz’s crossing-number conjecture, J. Graph Theory 17 (1993) 657–671. [26] W.P. Zheng, X.H. Lin, Y.S. Yang, C. Cui, On the crossing number of Km Pn , Graphs Combin. 23 (2007) 327–336.