Remarks on the Thickness of Kn,N,N 1 Introduction 2 the Proofs Of

∗ Remarks on the Thickness of Kn,n,n

Yan Yang Department of Mathematics Tianjin University, Tianjin 300072, P.R.China [email protected]

Abstract The thickness θ(G) of a graph G is the minimum number of planar subgraphs into which G can be decomposed. In this paper, we provide a new upper bound for the n+1 thickness of the complete tripartite graphs Kn,n,n (n ≥ 3) and obtain θ(Kn,n,n) = 3 , when n ≡ 3 (mod 6). Keywords thickness; complete tripartite graph; planar subgraphs decomposition. Mathematics Subject Classiﬁcation 05C10.

1 Introduction The thickness θ(G) of a graph G is the minimum number of planar subgraphs into which G can be decomposed. It was deﬁned by Tutte [10] in 1963, derived from early work on biplanar graphs [2,11]. It is a classical topological invariant of a graph and also has many applications to VLSI design, graph drawing, etc. Determining the thickness of a graph is NP-hard [6], so the results about thickness are few. The only types of graphs whose thicknesses have been determined are complete graphs [1,3], complete bipartite graphs [4] and hypercubes [5]. The reader is referred to [7,8] for more background on the thickness problems.

In this paper, we study the thickness of complete tripartite graphs Kn,n,n,(n ≥ 3). When n = 1, 2, it is easy to see that K1,1,1 and K2,2,2 are planar graphs, so the thickness n of both ones is one. Poranen proved θ(Kn,n,n) ≤ 2 in [9] which was the only result about the thickness of Kn,n,n, as far as the author knows. We will give a new upper bound for θ(Kn,n,n) and provide the exact number for the thickness of Kn,n,n, when n is congruent to 3 mod 6, the main results of this paper are the following theorems.

n+1 Theorem 1. For n ≥ 3, θ(Kn,n,n) ≤ 3 + 1. n+1 Theorem 2. θ(Kn,n,n) = 3 when n ≡ 3 (mod 6).

2 The proofs of the theorems

∗Supported by the National Natural Science Foundation of China under Grant No. 11401430 2

In [4], Beineke, Harary and Moon determined the thickness of complete bipartite graph

Km,n for almost all values of m and n.

mn Lemma 3.[4] The thickness of Km,n is 2(m+n−2) except possibly when m and n are 2k(m−2) odd, m ≤ n and there exists an integer k satisfying n = m−2k .

n+1 Lemma 4. For n ≥ 3, θ(Kn,n,n) ≥ 3 .

Proof. Since Kn,2n is a subgraph of Kn,n,n, we have θ(Kn,n,n) ≥ θ(Kn,2n). From n+1 Lemma 3, the thickness of Kn,2n (n ≥ 3) is 3 , so the lemma follows.

For the complete tripartite graph Kn,n,n with the vertex partition (A, B, C), where A = {a0, . . . , an−1}, B = {b0, . . . , bn−1} and C = {c0, . . . , cn−1}, we deﬁne a type of graphs, they are planar spanning subgraphs of Kn,n,n, denoted by G[aibj+ick+i], in which 0 ≤ i, j, k ≤ n−1 and all subscripts are taken modulo n. The graph G[aibj+ick+i] consists of n triangles aibj+ick+i for 0 ≤ i ≤ n − 1 and six paths of length n − 1, they are

a0bj+1ck+2a3bj+4ck+5 . . . a3ibj+3i+1ck+3i+2 ...,

cka1bj+2ck+3a4bj+5 . . . ck+3ia3i+1bj+3i+2 ...,

bjck+1a2bj+3ck+4a5 . . . bj+3ick+3i+1a3i+2 ...,

a0ck+1bj+2a3ck+4bj+5 . . . a3ick+3i+1bj+3i+2 ...,

bja1ck+2bj+3a4ck+5 . . . bj+3ia3i+1ck+3i+2 ...,

ckbj+1a2ck+3bj+4a5 . . . ck+3ibj+3i+1a3i+2 ....

Equivalently, the graph G[aibj+ick+i] is the graph with the same vertex set as Kn,n,n and edge set

{aibj+i−1, aibj+i, aibj+i+1, aick+i−1, aick+i, aick+i+1 | 1 ≤ i ≤ n − 2}

∪{bj+ick+i−1, bj+ick+i, bj+ick+i+1 | 1 ≤ i ≤ n − 2}

∪{a0bj, a0bj+1, an−1bj+n−2, an−1bj+n−1}

∪{a0ck, a0ck+1, an−1ck+n−2, an−1ck+n−1)}

∪{bjck, bjck+1, bj+n−1ck+n−2, bj+n−1ck+n−1}.

Figure 1(a) illustrates the planar spanning subgraph G[aibici] of K5,5,5. 3

c0 b0 a1 c1

b2 a2 c3 b3 a4 c4

(a) The subgraph G1 = G[aibici] of K5,5,5

b0 a0 c c 2 3 c2 c3 a2 a3

b4 b1 a4 a1 b0

a1 a4 c1 c4 c1 c4

a0 b3 b2 a3 a2 c c0 0

(b) The subgraph G2 of K5,5,5 (c) The subgraph G3 of K5,5,5

Figure 1 A planar subgraphs decomposition of K5,5,5

Theorem 5. When n = 3p + 2 (p is a positive integer), θ(Kn,n,n) ≤ p + 2. Proof. When n = 3p+2 (p is a positive integer), we will construct two diﬀerent planar subgraphs decompositions of Kn,n,n according to p is odd or even, in which the number of planar subgraphs is p + 2 in both cases.

Case 1. p is odd. Let G1,...,Gp be p planar subgraphs of Kn,n,n where p+1 Gt = G[aibi+3(t−1)ci+6(t−1)], for 1 ≤ t ≤ 2 ; and Gt = G[aibi+3(t−1)ci+6(t−1)+2], for p+3 2 ≤ t ≤ p and p ≥ 3. From the structure of G[aibj+ick+i], we get that no two edges in G1,...,Gp are repeated. Because subscripts in Gt, 1 ≤ t ≤ p are taken modulo n, {3(t − 1) (mod n) | 1 ≤ t ≤ p} = {0, 3, 6,..., 3(p − 1)}, {6(t − 1) (mod n) | 1 ≤ t ≤ p+1 p+3 2 } = {0, 6,..., 3(p−1)} and {6(t−1)+2 (mod n) | 2 ≤ t ≤ p} = {3, 9,..., 3(p−2)}, the subscript sets of b and c in Gt, 1 ≤ t ≤ p are the same, i.e., 4

{i + 3(t − 1) (mod n) | 1 ≤ t ≤ p}

p + 1 p + 3 = {i + 6(t − 1) (mod n) | 1 ≤ t ≤ } ∪ {i + 6(t − 1) + 2 (mod n) | ≤ t ≤ p}. 2 2

Furthermore, if there exists t ∈ {1, . . . , p} such that aibj is an edge in Gt, then aicj is an edge in Gk for some k ∈ {1, . . . , p}. If the edge aibj is not in any Gt, then neither is the edge aicj in any Gt, for 1 ≤ t ≤ p.

From the construction of Gt, the edges that belong to Kn,n,n but not to any Gt, 1 ≤ t ≤ p, are

a0b3(t−1)−1, a0c3(t−1)−1, 1 ≤ t ≤ p (1)

an−1b3(t−1), an−1c3(t−1), 1 ≤ t ≤ p (2)

aibi−3, aibi−2, 0 ≤ i ≤ n − 1 (3)

aici−3, aici−2, 0 ≤ i ≤ n − 1 (4) p + 3 b c , b c , 0 ≤ i ≤ n − 1 and t = (5) i i+3(t−1)−1 i i+3(t−1) 2 p + 1 b c , b c , 1 ≤ t ≤ (6) 3(t−1) 6(t−1)−1 3(t−1)−1 6(t−1) 2 p + 3 b c , b c , ≤ t ≤ p and p ≥ 3 (7) 3(t−1) 6(t−1)+1 3(t−1)−1 6(t−1)+2 2

Let Gp+1 be the graph whose edge set consists of the edges in (3) and (5), and Gp+2 be the graph whose edge set consists of the edges in (1), (2), (4), (6) and (7). In the following, we will describe plane drawings of Gp+1 and Gp+2. (a) A planar embedding of Gp+1.

Place vertices b0, b1, . . . , bn−1 on a circle, place vertices ai+3 and c n+1 in the middle of i+ 2 bi and bi+1, join each of ai+3 and c n+1 to both bi and bi+1, we get a planar embedding i+ 2 of Gp+1. For example, when p = 1, n = 5, Figure 1(b) shows the subgraph G2 of K5,5,5. (b) A planar embedding of Gp+2. Firstly, we place vertices c0, c1, . . . , cn−1 on a circle, join vertex ai+3 to ci and ci+1, for

0 ≤ i ≤ n−1 , so that we get a cycle of length 2n. Secondly, join vertex an−1 to c3(t−1) for 1 ≤ t ≤ p, with lines inside of the cycle. Let `t be the line drawn inside the cycle joining p+1 p+3 an−1 with c6(t−1)−1 if 1 ≤ t ≤ 2 or with c6(t−1)+1 if 2 ≤ t ≤ p (p ≥ 3). For 1 ≤ t ≤ p, insert the vertex b3(t−1) in the line `t. Thirdly, join vertex a0 to c3(t−1)−1 for 1 ≤ t ≤ p, 0 with lines outside of the cycle. Let `t be the line drawn outside the cycle joining a0 with p+1 p+3 c6(t−1) if 1 ≤ t ≤ 2 or with c6(t−1)+2 if 2 ≤ t ≤ p (p ≥ 3). For 1 ≤ t ≤ p, insert the 0 vertex b3(t−1)−1 in the line `t. In this way, we can get a planar embedding of Gp+2. For example, when p = 1, n = 5, Figure 1(c) shows the subgraph G3 of K5,5,5. Summarizing, when p is an odd positive integer and n = 3p+2, we get a decomposition of Kn,n,n into p + 2 planar subgraphs G1,...,Gp+2. 5

Case 2. p is even. Let G1,...,Gp be p planar subgraphs of Kn,n,n where p Gt = G[aibi+3(t−1)ci+6(t−1)+3], for 1 ≤ t ≤ 2 ; and Gt = G[aibi+3(t−1)ci+6(t−1)+2], for p+2 2 ≤ t ≤ p. With a similar argument to the proof of Case 1, we can get that the subscript sets of b and c in Gt, 1 ≤ t ≤ p are the same, i.e., {i + 3(t − 1) (mod n) | 1 ≤ t ≤ p} p p + 2 = {i + 6(t − 1) + 3 (mod n) | 1 ≤ t ≤ } ∪ {i + 6(t − 1) + 2 (mod n) | ≤ t ≤ p}. 2 2

From the construction of Gt, G p and G p+2 have n − 2 edges in common, they are 2 2 b p+2 c p+2 , 1 ≤ i ≤ n − 1 and i 6= n − 4, we can delete them in one of these i+3( 2 −1) i+6( 2 −1)+1 two graphs to avoid repetition.

The edges that belong to Kn,n,n but not to any Gt, 1 ≤ t ≤ p, are

a0b3(t−1)−1, a0c3(t−1)−1, 1 ≤ t ≤ p (8)

an−1b3(t−1), an−1c3(t−1), 1 ≤ t ≤ p (9)

aibi−3, aibi−2, 0 ≤ i ≤ n − 1 (10)

aici−3, aici−2, 0 ≤ i ≤ n − 1 (11)

bici−1, bici, bici+1, 0 ≤ i ≤ n − 1 (12) p b c , 1 ≤ t ≤ (13) 3(t−1) 6t−4 2 p + 2 b c , < t ≤ p (14) 3(t−1) 6t−5 2 p b c , 1 ≤ t < (15) 3(t−1)−1 6t−3 2 p + 2 b c , ≤ t ≤ p (16) 3(t−1)−1 6t−4 2 Let Gp+1 be the graph whose edge set consists of the edges in (10), (11) and (12), and Gp+2 be the graph whose edge set consists of the edges in (8), (9), (13), (14), (15) and (16). We draw Gp+1 in the following way. Firstly, place vertices b0, c0, b1, c1, . . . , bn−1, cn−1 on a circle C, join vertex ci to bi and bi+1, we get a cycle of length 2n. Secondly, place 0 vertices a0, a2, . . . , an−2 on a circle C in the unbounded region deﬁned by the circle C 0 such that C is contained in the closed disk deﬁned by C , place vertices a1, a3, . . . , an−1 00 on a circle C contained in the bounded region of C. Join ai to bi−3, bi−2, ci−3, and ci−2, join bi to ci+1. We can get a planar embedding of Gp+1, so it is a planar graph. Gp+2 is also planar because it is a subgraph of a graph homeomorphic to a dipole (two vertices joined by some edges). For example, when p = 2, n = 8, Figure 2(c) and Figure 2(d) show the subgraphs G3 and G4 of K8,8,8 respectively. Summarizing, when p is an even positive integer and n = 3p + 2, we obtain a decomposition of Kn,n,n into p + 2 planar subgraphs G1,...,Gp+2. 6

Theorem follows from Cases 1 and 2.

From the proof of Theorem 5, we draw planar subgraphs decompositions of K5,5,5 and K8,8,8 as illustrated in Figure 1 and Figure 2 respectively.

c3 b0 a1 c4 b2 a2 c6 b3 a4 c7 b5 a5 c1 b6 a7 c2

(a) The subgraph G1 = G[aibici+3] of K8,8,8

c0 b3 a1 c1 b5 a2 c3 b6 a4 c4 b0 a5 c6 b1 a7 c7

(b) The subgraph G2 − b4c0 − b5c1 − b6c2 − b0c4 − b1c5 − b2c6 of K8,8,8 in which G2 = G[aibi+3ci] 7

b0 a0 c7 c0

b7 b1 a3 c6 c1 a1 a 4 b7 b2 c2 c7

b6 b2

a0 a c5 5 c2 a7 c3 c0 b0 b3 b5 b3 c4 c3 b4

a7 a6

Figure 2 A planar subgraphs decomposition of K8,8,8

Proof of Theorem 1. Because Kn−1,n−1,n−1 is a subgraph of Kn,n,n, θ(Kn−1,n−1,n−1) ≤ θ(Kn,n,n), by Theorem 5, θ(Kn,n,n) ≤ p + 2 also holds, when n = 3p or n = 3p + 1 (p is a positive integer), the theorem follows. Proof of Theorem 2. When n = 3p is odd, i.e., n ≡ 3 (mod 6), we decompose

Kn,n,n into p + 1 planar subgraphs G1,...,Gp+1, where Gt = G[aibi+3(t−1)ci+6(t−1)], for 1 ≤ t ≤ p. With a similar argument to the proof of Theorem 5, we can get that the subscript sets of b and c in Gt, 1 ≤ t ≤ p are the same, i.e.,

{i + 3(t − 1) (mod n) | 1 ≤ t ≤ p} = {i + 6(t − 1) (mod n) | 1 ≤ t ≤ p}.

If the edge aibj is in Gt for some t ∈ {1, . . . , p}, then there exists k ∈ {1, . . . , p} such that aicj is in Gk. If the edge aibj is not in any Gt, then neither is the edge aicj in any Gt, for 1 ≤ t ≤ p.

From the construction of Gt = G[aibi+3(t−1)ci+6(t−1)], we list the edges that belong to Kn,n,n but not to any Gt, 1 ≤ t ≤ p, as follows.

a0b3(t−1)−1, a0c6(t−1)−1, 1 ≤ t ≤ p (17)

an−1b3(t−1), an−1c6(t−1), 1 ≤ t ≤ p (18)

b3(t−1)c6(t−1)−1, b3(t−1)−1c6(t−1), 1 ≤ t ≤ p (19)

Let Gp+1 be the graph whose edge set consists of the edges in (17), (18) and (19). It is easy to see that Gp+1 is homeomorphic to a dipole and it is a planar graph. Summarizing, when p is an odd positive integer and n = 3p, we obtain a decomposition of Kn,n,n into p+1 planar subgraphs G1,...,Gp+1, therefor θ(Kn,n,n) ≤ p+1. Combining this fact and Lemma 4, the theorem follows. 8

According to the proof of Theorem 2, we draw a planar subgraphs decomposition of

K3,3,3 as shown in Figure 3.

c2 a0

b1 b2 c2

c0 b0 c0 b0

a1 c1

b2 a2 a2

(a) The subgraph G1 = G[aibici] of K3,3,3 (b) The subgraph G2 of K3,3,3

Figure 3 A planar subgraphs decomposition of K3,3,3

For some other θ(Kn,n,n) with small n, combining Lemma 4 and Poranen’s result men- tioned in Section 1, we have θ(K4,4,4) = 2, θ(K6,6,6) = 3. Since there exists a decomposition of K7,7,7 with three planar subgraphs as shown in Figure 4, Lemma 4 implies that n+1 θ(K7,7,7) = 3. We also conjecture that the thickness of Kn,n,n is 3 for all n ≥ 3.

c0 b0 a1 c1

b2 a2 c3 b3 a4 c4

b5 a5 c6 b6 (a) The subgraph G1 = G[aibici] of K7,7,7 9

c4 b2 a1 c5

b4 a2 c0 b5 a4 c1

b0 a5 c3 b1

(b) The subgraph G2 − a1b2 − a2b3 − a3b4 − a4b5 − a5b6 − b0c1 − b1c2 − b3c4 − b4c5 − b5c6 of K7,7,7 in which G2 = G[aibi+2ci+4]

b0 b2 b4 b6

a2 c4 a3 c5 c6 a4 c0 a5 c1 a6 c2 a0 c3 a1

b1 b3 b5

Figure 4 A planar subgraphs decomposition of K7,7,7 10

Acknowledgements The author is grateful to the anonymous referees for many helpful comments and sug- gestions.

References

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