Complete Bipartite Graphs

Let K_r,s denote a complete bipartite graph with partite sets of sizes r and s. If r = s = n, then such a graph is called a balanced complete bipartite graph and denoted Kn,n. Without loss of generality, we assume that s ≥ r.

In Chapter 2, we had mentioned that the following result by Fu and Huang [10]

about the linear 2-arboricity of K_n,n. Theorem 4.1.1. la₂(K_n,n) =

»

n²

b⁴ⁿ3 c

¼ .

Naturally, we would like to determine the linear 2-arboricity of K_r,s when s > r.

So, we begin with the case s ≥ 2r of K_r,s. Theorem 4.1.2. If s ≥ 2r, then la₂(K_r,s) =§_s

2

¨.

Proof. Assume that the partite sets of Kr,s are X = {x0, x1, . . . , xr−1} and Y = {y0, y1, . . . , ys−1}. For 0 ≤ j ≤ d^s₂e − 1, we define Nj as the set {y2j, y2j+1} except N_d2^s_e−1= {y_s−1} when s is odd. Then K_r,s can be viewed as K_r,d^s_2e with nodes x_i, N_j and unordered pairs of nodes (x_i, N_j) for 0 ≤ i ≤ r − 1 and 0 ≤ j ≤ d^s₂e − 1.

Moreover, since each unordered pair of nodes (x_i, N_j) in K_r,d^s_2e is composed of a path y_2jx_iy_2j+1 in original K_r,s except (x_i, N_d^s_2e−1) which is an edge x_iy_s−1 when s is odd. Therefore, for ` = 0, 1, . . . , d<sup>s</sup><sub>2</sub>e − 1, the unordered pairs of nodes with bipartite difference ` in K_r,d^s_2e can produce a linear 2-forest in original K_r,s. Hence, la₂(K_r,s) ≤ d^s₂e. On the other hand, if s ≥ 2r, then it is not difficult to see that a linear 2-forest in Kr,shas at most 2r edges and then la2(Kr,s) ≥ d^|E(K_2r^r,s)|e = d^rs_2re = d^s₂e.

In what follows, we consider the cases when 2r ≥ s > r. First, let P_n be a path with n vertices. An earlier work of Ushio [25] had shown the following:

Theorem 4.1.3. K_r,s has a P₃-factorization if and only if (i) r + s ≡ 0 (mod 3), (ii) 2s ≥ r, (iii) 2r ≥ s, and (iv) _2(r+s)^3rs is an integer.

Recall that if a graph G has a P_k-factorization then la_k−1(G) is equal to (k−1)·|V (G)|^k·|E(G)| . Thus, by Theorem 4.1.3, we have the following corollary:

Corollary 4.1.4. Assume 2r ≥ s ≥ r. If r + s ≡ 0 (mod 3) and _2(r+s)^3rs is an integer, then la₂(K_r,s) = _2(r+s)^3rs .

Next, we assume that the graph Kr,s does not have a P3-factorization and let s = 2r − t. Then r > t ≥ 0, i.e., r ≥ t + 1.

Proposition 4.1.5. If 3` + 2 ≥ t ≥ 3` and r ≥ λt + 1, then la₂(K_r,2r−t) ≥ r − ` + dλ(3`+2)−2`−1

2λ(3`+2)−2` · `e.

Proof. If 3` + 2 ≥ t ≥ 3`, then b^{2|V (K}₃^r,2r−t)|c = b^2(3r−t)₃ c = b2r −^2t₃c = b2r − t +₃^tc = 2r −t+`. By Lemma 2.1.5, la₂(K_r,2r−t) ≥ d ^{|E(Kr,2r−t)|}

b2|V (Kr,2r−t)|

3 ce = dr·(2r−t)2r−t+`</sub>e = dr −<sub>2r−t+`r`</sup> e = dr − ` + (2r−t+`</sub><sup>r−t+` · `)e = r − ` + d2r−t+`</sub><sup>r−t+`</sup> · `e. Since 3` + 2 ≥ t ≥ 3` and r ≥ λt + 1, we have <sub>2r−t+`<sup>r−t+`</sup> ≥ <sub>2λt+2−t+`<sup>λt+1−t+`</sup> = (2λ−1)t+`+2<sup>(λ−1)t+`+1</sup> ≥ (λ−1)(3`+2)+`+1

(2λ−1)(3`+2)+`+2 = λ(3`+2)−2`−1

2λ(3`+2)−2` . Hence, la₂(K_r,2r−t) ≥ r − ` + dλ(3`+2)−2`−1

2λ(3`+2)−2` · `e.

Corollary 4.1.6. If 3` + 2 ≥ t ≥ 3` and r ≥ t + 1, then la2(Kr,2r−t) ≥ r − ` + d<sub>4</sub><sup>`</sup>e.

Proof. From Proposition 4.1.5 and let λ = 1, then λ(3`+2)−2`−1

2λ(3`+2)−2` = <sub>4(`+1)</sub><sup>`+1</sup> = ¹₄. Finally, we conclude the work of this section with the following theorem.

Theorem 4.1.7. If 5 ≥ t ≥ 0 and r ≥ t + 1, then la₂(K_r,2r−t) = r.

Proof. From Corollary 4.1.6 and let ` = 0, 1, then we have la₂(K_r,2r−t) ≥ r. On the other hand, by Theorem 4.1.2, we know that la2(Kr,2r) = r. Thus, la2(Kr,2r−t) ≤ la2(Kr,2r) = r.

4.2 Complete Graphs

In Chapter 2, we had mentioned the following result by Chen et al. [3] about the linear 2-arboricity of a complete graph Km.

Proposition 4.2.1. la₂(K_12t+11) = 9t + 9 for any t ≥ 0.

However, the answer 9t + 9 of la2(K12t+11) is wrong, because some computing errors happened in its proof. Hence, in this section, we will give a revised result that la₂(K_12t+10) = la₂(K_12t+11) = 9t + 8 for any t 6= 4. Moreover, this result also solve a problem raised by Bermond et al. [2] almost completely.

Before we go any further, we need some more definitions. Let S = {1, 2, . . . , ν}

be a set of ν elements. A latin square of order ν is a ν × ν array in which each cell contains a single element from S, such that each element occurs exactly once in each row and exactly once in each column. If in a latin square L of order ν the r² cells defined by r rows and r columns form a latin square of order r it is a latin subsquare of L. A latin square L = [`ij] is said to be symmetric if `ij = `ji for all 1 ≤ i, j ≤ ν.

An incomplete latin square ILS(ν; b₁, b₂, . . . , b_κ) is a ν × ν array A with entries from a set B of size ν, where B_i ⊆ B for 1 ≤ i ≤ κ with |B_i| = b_i, and B_i∩ B_j = ∅ for 1 ≤ i, j ≤ κ. Moreover,

1. each cell of A is empty or contains an element of B;

2. the subarrays indexed by B_i× B_i are empty (these subarrays are holes); and 3. the elements in row or column b are exactly those of B − B_i if b ∈ B_i, and of B otherwise.

A partitioned complete latin square PILS(ν; b1, b2, . . . , bκ) is an incomplete latin square with b₁ + b₂ + · · · + b_κ = ν. Figure 4.1 is an example of a symmetric PILS(8; 2, 2, 2, 2).

8 6 7 3 4 5 5 7 4 8 3 6

8 5 1 7 6 2

6 7 8 2 5 1

7 4 1 8 2 3

3 8 7 2 1 4

4 3 6 5 2 1 5 6 2 1 3 4

Figure 4.1: An example of a symmetric PILS(8; 2, 2, 2, 2).

It is worthy of noting that, in 1987, Fu [9] proved that:

Theorem 4.2.2. A symmetric partitioned complete latin square PILS(2κ; 2, 2, . . . , 2) exists for each κ ≥ 3.

Next, we want to show some lemmas. For convenience, the vertices in K_m are denoted v0, v1, . . . , vm−1.

Lemma 4.2.3. la2(K11) = 8.

Proof. We construct the array in Figure 4.2 to show that la₂(K₁₁) ≤ 8. The entry ω in row v_γ and column v_δ means that the edge v_γv_δ appears in the linear 2-forest labelled by ω. On the other hand, by Lemma 2.1.5, la₂(K₁₁) ≥ d_b2·11⁵⁵

3 ce = 8.

Lemma 4.2.4. la₂(K₁₂− M) = 8 where M is a matching of size 3 in K₁₂.

Proof. Without loss of generality, let the matching M be the set {v₁v₄, v₆v₁₀, v₇v₁₁} in K₁₂. Then the array in Figure 4.3 shows that la₂(K₁₂− M) ≤ 8. On the other hand, by Lemma 2.1.5, la₂(K₁₂− M) ≥ d_b2·12⁶³

3 ce = 8.

Lemma 4.2.5. la2(K35) = 26.

Proof. The array in Figure 4.4 shows that la2(K35) ≤ 26. Since it is symmetric, we omit the entries of half the array. On the other hand, by Lemma 2.1.5, la₂(K₃₅) ≥ d_b2·35⁵⁹⁵

3 ce = 26.

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9v10 v11 v12 v13 v14 v15 v16 v17v18 v19 v20 v21v22v23

Figure 4.4: The symmetric array shows that la2(K35) ≤ 26.

Lemma 4.2.6. la₂(K_12,12) = 9.

Now, we are ready to obtain the main results.

Proposition 4.2.8. la2(K12t+11) = 9t + 8 for any t ≥ 0 and t is odd.

Proof. First, we partition the vertex set of K12t+11 into t + 1 disjoint subsets S₀, S₁, . . . , S_t, where S_i = {v_i[0], v_i[1], . . . , v_i[11]} for all i = 0, 1, . . . , t − 1 and S_t = {x₀, x₁, . . . , x₁₀}. Hence, the subgraph of K_12t+11 induced by S_i is a K₁₂ or a K₁₁, for

v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 Lemma 3.1.3, K_12t+11 can be decomposed into t K_12,12-factors in each of which there exists one component is K_11,12, and a K₁₂-factor in which there exists one component is K₁₁. Then, for the K₁₂-factor, the edges of K₁₂− {v_i[1]v_i[4], v_i[6]v_i[10], v_i[7]v_i[11]} for 0 ≤ i ≤ t−1 in those K₁₂and the edges of K₁₁can produce eight linear 2-forests from Lemmas 4.2.3 and 4.2.4. Moreover, since the edges v_i[1]v_i[4], v_i[6]v_i[10], v_i[7]v_i[11] in each K₁₂ of the K₁₂-factor are not being used, then we unite them with the corresponding component K11,12 of each K12,12-factor. Hence, for each K12,12-factor, the edges in K11,12∪G[{vi[1]vi[4], vi[6]vi[10], vi[7]vi[11]}] and the edges in those K12,12 can produce nine linear 2-forests from Lemmas 4.2.6 and 4.2.7. Therefore, from the t K_12,12-factors and a K₁₂-factor of K_12t+11, we have that la₂(K_12t+11) ≤ 9t + 8. On the other hand, by Lemma 2.1.5, la₂(K_12t+11) ≥ d(12t+11)(12t+10)

2b^2(12t+11)₃ c e = 9t + 8. This concludes the proof.

Proposition 4.2.9. la₂(K_12t+11) = 9t + 8 for any t ≥ 6 and t is even.

Proof. We prove this proposition by using the techniques on latin squares proposed by Chen et al. [3]. First, let the 35 × 35 array in Figure 4.4 be partitioned into four subarrays P, Q, Q^T, R as shown in Figure 4.7, where P, Q, and R are 24 × 24, 24 × 11, and 11 × 11 arrays respectively. Moreover, let the 12 × 12 array in Figure 4.5 also be denoted W .

Q^T R

P Q

Figure 4.7: Four subarrays of the array in Figure 4.4.

Next, since t ≥ 6 and t is even, from Theorem 4.2.2, we can find a symmetric PILS(2κ; 2, 2, . . . , 2) such that t = 2κ. We use L = [`_ij] to denote this symmetric PILS(2κ; 2, 2, . . . , 2). Then, from L, we can construct a (12t + 11) × (12t + 11) symmetric array L⁰ as shown in Figure 4.8 to show that la₂(K_12t+11) ≤ 9t + 8, where 1. B_x is a 24 × 24 array, for 1 ≤ x ≤ κ;

2. the entry B_x(r, s) in B_x equals P (r, s) in P if P (r, s) ∈ {1, 2, . . . , 8}, for 1 ≤ x ≤ κ;

3. B_x(r, s) = P (r, s) + (x − 1) · 18 if P (r, s) 6∈ {1, 2, . . . , 8}, for 1 ≤ x ≤ κ;

4. the 12 × 12 array Cij = W + 8 + (`ij − 1) · 9, for 1 ≤ i, j ≤ 2κ;

5. the 24 × 11 array Dx = Q + (x − 1) · 18, for 1 ≤ x ≤ κ; and 6. the 11 × 11 array E = R.

On the other hand, by Lemma 2.1.5, la₂(K_12t+11) ≥ d(12t+11)(12t+10)

2b^2(12t+11)₃ c e = 9t + 8.

Theorem 4.2.10. la₂(K_12t+10) = la₂(K_12t+11) = 9t + 8 for any t 6= 4.

Proof. By Lemma 4.2.5 and Propositions 4.2.8 ∼ 4.2.9, la₂(K_12t+11) = 9t + 8 for any t 6= 4. Moreover, 9t + 8 = la₂(K_12t+11) ≥ la₂(K_12t+10) ≥ d(12t+10)(12t+9)

2b^2(12t+10)₃ c e = 9t + 8.

B₁

E B₂

B_k D₁

D₂

D_k

D₂^T

D₁^T D_k^T

C_ij

C_ij^T

Figure 4.8: A (12t + 11) × (12t + 11) symmetric array.

4.3 Balanced Complete Multipartite Graphs

In 1989, Ushio and Tsuruno [26] showed the following result on balanced complete multipartite graphs K_m(n).

Theorem 4.3.1. K_m(n) has a P₃-factorization if and only if mn ≡ 0 (mod 3) and (m − 1)n ≡ 0 (mod 4).

Recall that if a graph G has a P_k-factorization then la_k−1(G) is equal to (k−1)·|V (G)|^k·|E(G)| . Thus, by Theorem 4.3.1, we have the following corollary:

Corollary 4.3.2. la2(Km(n)) = ^3(m−1)n₄ when mn ≡ 0 (mod 3) and (m − 1)n ≡ 0 (mod 4).

In what follows, we consider the cases when K_m(n)does not have a P₃-factorization and begin with the case m = 3 of K_m(n).

Lemma 4.3.3. la₂(K_3(n)) = §_3n

2

¨ if n ≡ 1 (mod 2).

Proof. Assume that the partite sets of K_3(n) are V0 = {v_0[0], v_0[1], . . . , v_0[n−1]}, V1 = {v1[0], v1[1], . . . , v1[n−1]}, and V2 = {v2[0], v2[1], . . . , v2[n−1]}. First, for all 0 ≤ α 6= β ≤ 2, let the balanced complete bipartite subgraph of K_3(n)induced by V_αand V_β be denoted G(V_α, V_β).

Then, for any ² ∈ {0, 1, ..., n − 2}, we observe that the edges with bipartite differences ², ² + 1 in all of G(V₁, V₂), G(V₂, V₃), and G(V₃, V₁) can produce three linear 2-forests. Hence, the edges with bipartite differences 1, 2, . . . , n − 1 in all of G(V₁, V₂), G(V₂, V₃), and G(V₃, V₁) can generate (ⁿ⁻¹₂ ) · 3 linear 2-forests, which are

©v_0[j]v_1[j+1+2r]v_2[j+2+4r]|j = 0, 1, . . . , n − 1ª ,©

v_2[j]v_0[j+1+2r]v_1[j+3+4r]|j = 0, . . . , n − 1ª , and ©

v1[j]v2[j+2+2r]v0[j+4+4r]|j = 0, 1, . . . , n − 1ª

for all r ∈ {0, 1, . . . ,ⁿ⁻¹₂ − 1}. Note that the index y of each vertex v_x[y] is modulo n. For example, Figure 4.9 shows that the edges with bipartite differences 1, 2 in all of G(V₁, V₂), G(V₂, V₃), and G(V₃, V₁) can produce three linear 2-forests in K₃₍₇₎.

v_0[6]

v_0[5]

v_0[4]

v_0[3]

v_0[2]

v_0[1]

v_0[0]

v_2[6]

v_2[5]

v_2[4]

v_2[3]

v_2[2]

v_2[1]

v_2[0]

v_1[6]

v_1[5]

v_1[4]

v_1[3]

v_1[2]

v_1[1]

v_1[0]

Figure 4.9: Three linear 2-forests in K₃₍₇₎.

Moreover, the disjoint 3-cycles induced by the edges with bipartite difference 0 in all of G(V₁, V₂), G(V₂, V₃), and G(V₃, V₁) can be decomposed into two linear 2-forests

©v_0[j]v_1[j]v_2[j]|j = 0, 1, . . . , n − 1ª and ©

v_2[j]v_0[j]|j = 0, 1, · · · , n − 1ª

. Thus, la₂(K_3(n))

≤ ³⁽ⁿ⁻¹⁾₂ + 2 = ³ⁿ⁺¹₂ =§_3n

2

¨ if n ≡ 1 (mod 2).

On the other hand, from Lemma 2.1.5, la₂(K_3(n)) ≥§_3n all 0 ≤ α 6= β ≤ m−1, let the balanced complete bipartite subgraph of K_m(n)induced by V_α and V_β be denoted G(V_α, V_β). Moreover, from Lemma 3.1.1 (by replacing each edge of K_mwith K_n,n), K_m(n)has a K_3(n)-factorization in which there are ^|E(Km)|

(^{|V (Km)|}3 )^·3 =

m−1

2 K3(n)-factors.

Then, from the proof of Lemma 4.3.3, we know that the edges with bipartite differences 1, 2, . . . , n − 1 in all of G(V_α, V_β) for 0 ≤ α 6= β ≤ m − 1 can generate

Proof. Dividing all m partite sets of Km(n) into ^m₂ disjoint pairs of two partite sets shows that K_m(n) is the union of K^m

Proof. Dividing all m partite sets of K_m(n) into ^m₂ disjoint pairs of two partite sets shows that K_m(n) is the union of K^m_2(2n) and one Kn,n-factor of K_m(n). Since

Proposition 4.3.7. la₂(K_m(n)) =

l3(m−1)n 4

m

if m ≡ 0 (mod 3) and n ≡ 0 (mod 2).

Proof. Dividing all m partite sets of K_m(n)into ^m₃ disjoint collections of three partite sets shows that Km(n) is the union of K^m₃(3n) and one K3(n)-factor of Km(n). Since 3n ≡ 0 (mod 6), from Corollary 4.3.2 and Propositions 4.3.5 ∼ 4.3.6, la2(Km(n)) ≤ la₂(K_3(n)) + la₂(K^m_3(3n)) = ³ⁿ₂ +

»

3(^m3−1)⁽³ⁿ⁾

4

¼

=

l3(m−1)n 4

m

. On the other hand, from Lemma 2.1.5, la2(Km(n)) ≥

l3(m−1)n 4

m

if m ≡ 0 (mod 3) and n ≡ 0 (mod 2).

Concluding Remark. The main goal of this section is to determine la₂(K_m(n)) when mn ≡ 0 (mod 3). However, we are not able to finish the whole part at this moment due to several stubborn subcases. We expect to settle the rest cases in the near future.

Chapter 5

在文檔中 完全多部圖之線性k蔭度 (頁 54-67)