The Thickness of Cartesian Product $ K N\Box P M$

$The Thickness of Cartesian Product $ K N\Box P M$$

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In this paper, the thickness of the cartesian product KnPm, n,m ≥ 2 is obtained, in which Kn is the complete graph with n vertices and Pm is the path with m vertices.

2 The lower bound for the thickness of KnPm, n, m ≥ 2 From Euler’s polyhedron formula, a planar graph G has at most 3|V |− 6 edges, furthermore, one can get the lower bound for the thickness of a graph as follows.

m Lemma 2.1[3] If G = (V, E) is a graph with |V | = n and |E| = m, then θ(G) ≥ 3n−6 .

As for the graph KnPm, the numbers of vertices and edges are |V | = mn and |E| = mn(n−1) 2 + n(m − 1) respectively. By using Lemma 2.1,

mn(n−1) 2 + n(m − 1) θ(KnPm) ≥ 3mn − 6 ln + 1 2 m = + 6 6(mn − 2) l m p +1 when n = 6p, 6p + 1, 6p + 2, 6p + 3, 6p + 4; = (1) ( p +2 when n = 6p + 5.

n+7 Lemma 2.2[2] The thickness of the complete graph Kn is θ(Kn)= 6 , except that θ(K9)= θ(K10) = 3.

3 The thickness of KnP2, n ≥ 2 1 2 Let Kn be the complete graph with n vertices labeled by v1, v2, . . . , vn respectively, Kn 1 is a copy of Kn and it’s vertices labeled by u1, u2, . . . , un respectively, by joining the vertices vi and ui by an edge viui, 1 ≤ i ≤ n, one can get the graph KnP2. Figure 1 illustrates K5P2.

By inserting a vertex wi at edge viui, for 1 ≤ i ≤ n, and merging these n 2-valent vertices w1, w2, . . . , wn into one vertex w, one can get a new graph. This graph also can 3 be seen as the vertex-amalgamation of Kn+1 and Kn+1 at w, denote by Kn+1 ∨{w} Kn+1. Figure 2 shows the graph K6 ∨{w} K6.

v1 u1

v5 v2 u2 u5

v4 v3 u3 u4

Figure 1 The graph K5P2

v1 u1

w u5 v5 v2 u2

v4 v3 u3 u4

Figure 2 The graph K6 ∨{w} K6

Lemma 3.1[7] If G is the vertex-amalgamation of G1 and G2, θ(G1)= n1 and θ(G2)= n2, then θ(G) = max{n1,n2}.

From Lemma 3.1, the thickness of Kn+1 ∨{w} Kn+1 is the same as the thickness of Kn+1. Let θ(Kn+1)= t and {G1, G2,...,Gt} is a planar decomposition of Kn+1, then one can get a planar decomposition of Kn+1 ∨{w} Kn+1 as follows,

{G1 ∨{w} G1, G2 ∨{w} G2,...,Gt ∨{w} Gt}, in which Gi ∨{w} Gi, 1 ≤ i ≤ t are plane graphs. In this way, a planar decomposition of

K6 ∨{w} K6 is shown in Figure 3.

v1 u1 v1 u1

w u5 v5 w u5 v5 v2 u2 v2 u2

v4 v4 v3 u3 u4 v3 u3 u4

Figure 3 A planar decomposition of K6 ∨{w} K6

From the construction of Gi∨{w}Gi, if the edge vqw ∈ Gi∨{w}Gi, then uqw ∈ Gi∨{w}Gi, 1 ≤ q ≤ n. For each graph Gi ∨{w} Gi, 1 ≤ i ≤ t, if vqw, uqw ∈ Gi ∨{w} Gi, then we 4 replace them by a new edge vquq, for 1 ≤ q ≤ n, and delete the vertex w. In this way, we can get a new planar decomposition, which is exactly a planar decomposition of KnP2. Figure 4 illustrates a planar decomposition of K5P2 by using this way.

v1 u1 v1 u1

u5 v5 w u5 v5 v2 u2 v2 u2

v4 v4 v3 u3 u4 v3 u3 u4

Figure 4 A planar decomposition of K5P2

From the argument and construction the above, one can get a planar decomposition of Kn P2 from that of Kn+1 ∨{w} Kn+1, so we have that θ(Kn P2) ≤ θ(Kn+1 ∨{w} Kn+1)= θ(Kn+1). (2)

Theorem 3.1 The thickness of the cartesian product KnP2(n ≥ 2) is θ(KnP2) = n+8 6 , except that θ(K8 P2) = θ(K9 P2) = 3 and possibly when n = 6p + 4 (n is a nonnegative integer). Proof When n 6= 8, 9, from (1), (2) and Lemma 2.2, we can get that θ(KnP2) = θ(Kn+1), except possibly when n = 6p + 4 (p is a nonnegative integer). When n = 8, we suppose that θ(K8P2) = 2, and {G1, G2} is a planar decomposition of K8P2 in which G1 and G2 are plane graphs. We insert 2-valent vertex wi at edge viui, for each 1 ≤ i ≤ n, in G1 and G2, then merge these 2-valent vertices into vertices w and w˜ in G1 and G2 respectively. In this way, we get two new plane graphs, denote by G˜1 and ˜ ˜ ˜ G2, and {G1, G2} is a planar decomposition of K9 ∨{w} K9. From (2), θ(K9)= θ(K9 ∨{w} K9) ≤ 2, it is a contradiction to the fact θ(K9) = 3. Hence, θ(K8P2) = θ(K9) = 3. With a similar argument, we can get θ(K9P2)= θ(K10) = 3. Summarizing the above, the theorem is obtained.

4 The thickness of KnPm, n ≥ 2, m ≥ 3 We use the similar method with that in section 3. Firstly, we insert a 2-valent vertex into each ”path edge” (the edges come from Pm). Secondly, we merge these (m − 1)n 2-valent vertices into m − 1 vertices, each of which joint two adjacent Kn, then we get a new graph G˜. The graph G˜ can be seen as a vertex-amalgamation of m graphs, in which the ﬁrst and the mth graphs are Kn+1, the others are Kn+2 − e. From Lemma 3.1, one can get that

θ(G˜)= θ(Kn+2 − e) ≤ θ(Kn+2). (3) 5

With a similar argument to that in section 3, we can get a planar decomposition of

KnPm from a planar decomposition of G˜, hence

θ(KnPm) ≤ θ(Kn+2 − e) ≤ θ(Kn+2). (4)

Theorem 4.1 The thickness of the cartesian product KnPm(n ≥ 2,m ≥ 3) is n+9 θ(Kn Pm)= 6 , except possibly when n = 6p+3, 6p+4 and n = 8 (p is a nonnegative integer). Proof When n 6= 7, 8, from (1), (4) and Lemma 2.2, we can get that θ(KnPm) =

θ(Kn+2), except possibly when n = 6p + 3, 6p + 4 (p is a nonnegative integer). When n = 7, from (1) and (4), we have 2 ≤ θ(K7Pm) ≤ θ(Kn+2 − e). We can give a planar decomposition of K9 − e as shown in Figure 5, and K9 − e is a non-planar graph, hence θ(K9 − e) = 2, θ(K7Pm) = 2. Summarizing the above, the theorem follows. 8 1 8

12343 567 3 5264

9 7 9

Figure 5 A planar decomposition of K9 − e

5 Concluded Remarks Determining the thickness for a arbitrary graph is a diﬃcult problem. The known results and methods are few. In [7], the authors ﬁrst considered getting the thickness of graphs by operations on graphs. In this paper, the author use operations on graphs and some

conclusions in [7], obtain the thickness of the cartesian product KnPm. n+8 For the exceptional cases in theorem 3.1, when n = 6p+4, θ(Kn P2) is either ⌊ 6 ⌋ or n+8 ⌊ 6 ⌋− 1. For the exceptional cases in theorem 4.1, when n = 6p + 3, 6p + 4, θ(Kn Pm) n+9 n+9 is either ⌊ 6 ⌋ or ⌊ 6 ⌋− 1; when n = 8, θ(K8 Pm) is either 2 or 3. What are the exactly numbers for these exceptional cases are still open.

References

[1] A. Aggarwal, M.Klawe, and P.Shor, Multilayer grid embeddings for VLSI, Algorithmica, 6(1991) 129-151 [2] L.W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math., 17 (1965) 850-859 [3] L.W. Beineke, F. Harary, and J.W. Moon, On the thickness of the complete bipartite graph, Proc. Cambridge. Philo. Soc., 60 (1964) 1-5 6

[4] M. Kleinert, Die Dicke des n-dimensionale Würfel-Graphen, J. Comb. Theory, 3 (1967) 10-15. [5] E. Mäkinen and T. Poranen. An annotated bibliography on the thickness, outerthickness, and arboricity of a graph. Technical report, University of Tampere, 2009. http://www.sis.uta.fi/ tp54752/pub/thickness-bibliography/. [6] W.T. Tutte, The thickness of a graph, Indag. Math., 25 (1963) 567-577 [7] Y. Yang, X.H. Kong. The thickness of amalgamations of graphs. http://arxiv.org/abs/1201.6483.