Balanced Complete Multipartite Graphs

In this section, we study the linear 3-arboricity of a balanced complete multipar-tite graph K_m(n)with mn ≡ 0 (mod 4). Before we go any further, we need some more lemmas.

Let Pα(β) be an α-partite graph such that each partite set Vi has β vertices for all i ∈ {0, 1, . . . , α − 1} and the edge uv ∈ E(P_α(β)) if and only if u ∈ V_w and v ∈ V_w+1 where w ∈ {0, 1, . . . , α − 2}.

Lemma 3.4.1. la_k(P_k+1(s)) = s.

Proof. First, for all i ∈ {0, 1, . . . , k}, assume that the vertices of partite set V_i in P_k+1(s) are v_i[0], v_i[1], . . . , v_i[s−1]. Then, let the `th linear k-forest be the set of P_k+1’s

©v_0[j]v<sub>1[j+(`−1)]</sub>. . . vk[j+k(`−1)]| j = 0, 1, . . . , s − 1ª

for all ` ∈ {1, 2, . . . , s}. Note that the index y of each vertex v_x[y] is modulo s. It is not difficult to check that the edges in the linear k-forests above are distinct and exactly all of the edges in P_k+1(s). Thus lak(Pk+1(s)) = s.

Lemma 3.4.2. la_k(K_m(sn)) ≤ s · la_k(K_m(n)).

Proof. We can obtain K_m(sn) from K_m(n) by replacing each edge of K_m(n) with K_s,s. Hence, a path P_λ in a linear k-forest of K_m(n) corresponds to a λ-partite subgraph P_λ(s) of K_m(sn), where 2 ≤ λ ≤ k + 1. Moreover, la_k(P_λ(s)) ≤ la_k(P_k+1(s)) for all 2 ≤ λ ≤ k + 1. Therefore, la_k(K_m(sn)) ≤ la_k(P_k+1(s)) · la_k(K_m(n)) = s · la_k(K_m(n)) by Lemmas 3.1.4 and 3.4.1.

Lemma 3.4.3. If n ≡ 0 (mod 2^σ) where σ ≥ 1, then K_m(n)has a K_2σⁿ_,2σⁿ-factorization and there are 2^σ(m − 1) K_2σⁿ_,2σⁿ-factors in it.

Proof. We prove this lemma by using induction on the number σ. Assume σ = 1.

From Lemma 3.1.3 (by replacing each edge of K2mwith Kⁿ₂,ⁿ₂), the graph K_2m(ⁿ2) has a Kⁿ₂,ⁿ₂-factorization in which there are 2m−1 Kⁿ₂,ⁿ₂-factors. Moreover, K_2m(ⁿ2) is the union of K_m(n) and a Kⁿ_2,ⁿ₂-factor of K_2m(ⁿ2). Hence, K^m(n) has a Kⁿ_2,ⁿ₂-factorization and there are 2m − 2 = 2(m − 1) Kⁿ_2,ⁿ₂-factors in it. This provides the basis.

For the induction step, suppose σ = h + 1 ≥ 2. The induction hypothesis is that K_m(n) has a Kⁿ

2h,_2hⁿ -factorization in which there are 2^h(m − 1) Kⁿ

2h,_2hⁿ-factors.

Since a Kⁿ

2h,_2hⁿ-factor can be decomposed into two K ⁿ

2h+1,_2h+1ⁿ -factors, then K_m(n) has a K ⁿ

2h+1,_2h+1ⁿ -factorization and there are 2·2^h(m−1) = 2^h+1(m−1) K ⁿ

2h+1,_2h+1ⁿ -factors in it. Therefore, by mathematical induction, the assertion holds.

Now, we are ready to prove the main results on la₃(K_m(n)).

Proposition 3.4.4. la₃(K_m(n)) ≤ ^2(m−1)n₃ if m ≡ 0 (mod 2) and n ≡ 0 (mod 6).

Proof. From Lemma 3.1.3 (by replacing each edge of K_m with K_n,n), K_m(n) has a K_n,n-factorization and there are m − 1 K_n,n-factors in it. Hence, la₃(K_m(n)) ≤ (m − 1) · la₃(K_n,n) = (m − 1) · ²ⁿ₃ = ^2(m−1)n₃ by Lemma 3.1.4 and Theorem 3.2.9.

Proposition 3.4.5. la3(K_m(n)) ≤ ^2(m−1)n₃ if n ≡ 0 (mod 12).

Proof. From Lemma 3.4.3, Km(n)has a Kⁿ₂,ⁿ₂-factorization in which there are 2m − 2 Kⁿ_2,ⁿ₂-factors. Therefore, la₃(K_m(n)) ≤ (2m−2)·la₃(Kⁿ_2,ⁿ₂) = (2m−2)·²(ⁿ2)

3 = ^2(m−1)n₃ by Lemma 3.1.4 and Theorem 3.2.9.

Proposition 3.4.6. la₃(K_m(n)) ≤ ^2(m−1)n₃ if m ≡ 4 (mod 12).

Proof. From Lemma 3.1.2, K_mhas a K₄-factorization and there are ^|E(Km)|

(^{|V (Km)|}4 )^·6 = ^m−1₃ K4-factors in it. Since la3(K4) = 2, from Lemmas 3.1.4 and 3.4.2, la3(K_m(n)) ≤ n · la3(Km) ≤ n · ^m−1₃ · la3(K4) = ^2(m−1)n₃ .

Proposition 3.4.7. la₃(K_m(n)) ≤ ^2(m−1)n₃ if m ≡ 1 (mod 3) and n ≡ 0 (mod 4).

Proof. Since 4m ≡ 4 (mod 12), from Lemma 3.1.2, K_4m has a K₄-factorization and there are ^³_{|V (K4m)|}^|E(K4m)|

4

´

·6 = ^4m−1₃ K₄-factors in it. Moreover, K_4m is the union of K_m(4) and one K₄-factor of K_4m. Hence, K_m(4) has a K₄-factorization in which there are ^4m−1₃ − 1 K₄-factors. By Lemmas 3.1.4 and 3.4.2, la₃(K_m(n)) ≤ ⁿ₄ · la₃(K_m(4)) ≤

n

4 ·¡_4m−1

3 − 1¢

· la3(K4) = ⁿ₄ · ^8(m−1)₃ = ^2(m−1)n₃ .

Proposition 3.4.8. la₃(K_m(n)) ≤ ^2(m−1)n₃ if m ≡ 0 (mod 4) and n ≡ 0 (mod 3).

Proof. Dividing all m partite sets of K_m(n) into ^m₄ disjoint collections of four partite sets shows that K_m(n)is the union of K^m_4(4n)and one K_4(n)-factor of K_m(n). Since 4n ≡ 0 (mod 12), by Propositions 3.4.5 and 3.4.6, la₃(K_m(n)) ≤ la₃(K_4(n)) + la₃(K^m_4(4n)) ≤

2(4−1)n

3 + ²(^m4−1)⁽⁴ⁿ⁾

3 = ^2(m−1)n₃ .

Proposition 3.4.9. la₃(K_m(n)) ≤ ^2(m−1)n₃ if m ≡ 10 (mod 12) and n ≡ 0 (mod 2).

Proof. From Lemma 3.1.3 (by replacing each edge of K_m with K_2,2), K_m(2) has a K_2,2-factorization and there are m − 1 K_2,2-factors in it. Moreover, since K_2,2 is consisting of a path P4 and one isolated edge, then a linear 3-forest can be induced by the set of P4’s in all K2,2 of any K2,2-factor in Km(2). Therefore, we obtain m − 1 linear 3-forests from the m − 1 K_2,2-factors of K_m(2). Now, we want to show that the isolated edges in those K_2,2 of K_2,2-factors in K_m(2) also produce linear 3-forests.

For all i ∈ {0, 1, . . . , m − 1}, let the vertices of partite set V_i in K_m(2) be denoted v_i[0] and v_i[1]. Without loss of generality, we assume that all isolated edges in those K_2,2of K_2,2-factors in K_m(2)are the edges of ^m₂ −1 perfect matchings U₁, U₂, . . . , U^m₂₋₁ and a matching M^m₂ in K_m(2), where U_` = ©

v_i[0]v_i+`[1]| i = 0, 1, . . . , m − 1ª

for ` ∈ {1, . . . ,^m₂ − 1} and M^m₂ =

n

vi[0]vi+^m₂[1]| i = 0, 2, . . . , m − 2 o

. Then, the edges of U₁, U₂, . . . , U^m₂₋₂ can generate ²(^m2−2)

3 linear 3-forests from the proof of Proposition 3.2.3 and the edges of U^m₂₋₁, M^m₂ also produce a linear 3-forest. Thus, la₃(K_m(n)) ≤

n

2 · la₃(K_m(2)) ≤ ⁿ₂ · [(m − 1) + ²(^m2−2)

3 + 1] = ^2(m−1)n₃ by Lemma 3.4.2.

Concluding the conditions of the pair (m, n) in the propositions given above, we find that mn ≡ 0 (mod 4) and (m − 1)n ≡ 0 (mod 3). On the other hand, by Lemma 2.1.5, it is easy to show that la₃(K_m(n)) ≥ ^2(m−1)n₃ if mn ≡ 0 (mod 4) and (m − 1)n ≡ 0 (mod 3). Therefore, we have the following:

Corollary 3.4.10. la₃(K_m(n)) = ^2(m−1)n₃ when mn ≡ 0 (mod 4) and (m − 1)n ≡ 0 (mod 3).

It is worthy of noting that, in 1999, Muthusamy and Paulraja [21] showed that:

Theorem 3.4.11. For k = p + 1 > 3 and p is a prime, K_m(n) has a P_k-factorization if and only if mn ≡ 0 (mod k) and 2(k − 1) | k(m − 1)n.

From the definitions of the linear (k − 1)-arboricity and a P_k-factorization of a graph, we know that if a graph G has a P_k-factorization then la_k−1(G) is equal to

k·|E(G)|

(k−1)·|V (G)|, which is the number of P_k-factors required to decompose G. Therefore, what we have proved gives an independent proof of the case k = 4 of Theorem 3.4.11.

Next, we consider the cases when K_m(n) does not have a P4-factorization.

Proposition 3.4.12. la3(Km(n)) ≤

l2(m−1)n 3

m

if m ≡ 0, 4, 6, 8 (mod 12) and n ≡ 4 (mod 6).

Proof. From Lemma 3.1.3 (by replacing each edge of K_m with K_n,n), K_m(n) has a K_n,n-factorization in which there are m − 1 K_n,n-factors. Hence, from the proof of Proposition 3.2.3, the edges with bipartite differences 1, 2, . . . , n − 1 in those K_n,n of K_n,n-factors in K_m(n) can generate (m − 1) · (²⁽ⁿ⁻¹⁾₃ ) linear 3-forests.

Moreover, it is not difficult to see that the subgraph induced by the set of edges with bipartite difference 0 in those Kn,n of Kn,n-factors in Km(n) is exactly a Km -factor. Therefore, by Theorem 3.3.9 and §_2m

3

¨ = §_2m−2

3

¨ if m ≡ 6 (mod 12), we have that la₃(K_m(n)) ≤ (m − 1) · (²⁽ⁿ⁻¹⁾₃ ) + la₃(K_m) = (m − 1) · (²⁽ⁿ⁻¹⁾₃ ) +§_2m−2

3

¨= l2(m−1)n

3

m .

Proposition 3.4.13. la₃(K_m(n)) ≤

l2(m−1)n 3

m

if m ≡ 2 (mod 6) and n ≡ 0 (mod 2).

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

m

2 ≡ 1 (mod 3) and 2n ≡ 0 (mod 4), by Theorem 3.2.9 and Proposition 3.4.7, la₃(K_m(n)) ≤ la₃(K_n,n) + la₃(K^m_2(2n)) ≤§_2n

3

¨+²(^m2−1)⁽²ⁿ⁾

3 =

l2(m−1)n 3

m . Proposition 3.4.14. la₃(K_m(n)) ≤

l2(m−1)n 3

m

if m ≡ 0 (mod 6) and n ≡ 2 (mod 6).

Proof. From Lemma 3.1.3 (by replacing each edge of K_m with K_n,n), K_m(n) has a K_n,n-factorization in which there are m − 1 K_n,n-factors. Hence, from the proof of Proposition 3.2.3, the edges with bipartite differences 2, 3, . . . , n − 1 in those K_n,n of Kn,n-factors in K_m(n) can generate (m − 1) · (²⁽ⁿ⁻²⁾₃ ) linear 3-forests.

Moreover, the edges with bipartite differences 0, 1 in those K_n,n of K_n,n-factors in K_m(n) also can produce m − 1 linear 3-forests except half of the edges with bipartite difference 1 in those K_n,n of K_n,n-factors in K_m(n) which are not being used. Thus, in what follows, we want to show that those edges which are not being used also produce linear 3-forests.

For all i ∈ {0, 1, . . . , m − 1}, let the vertices of partite set Vi in K_m(n) be denoted vi[0], vi[1], . . . , vi[n−1]. Without loss of generality, we assume that those edges which are not being used are the edges of ^m₂ − 1 perfect matchings U₁, U₂, . . . , U^m₂₋₁ and a matching M^m₂ in K_m(n), where

U_` =©

v_i[j]v<sub>i+`[j+1]</sub>| i ∈ {0, 1, . . . , m − 1}, j ∈ {1, 3, . . . , n − 1}ª for all ` ∈ {1, . . . ,^m₂ − 1} and

M^m₂ = n

v_i[j]v_i+^m_2[j+1]| i ∈ {0, 1, . . . ,^m₂ − 1}, j ∈ {1, 3, . . . , n − 1}

o .

Then, from the proof of Proposition 3.2.3, the edges of U₁, U₂, . . . , U^m₂₋₁, M^m₂ can generate ^2·₃^m2 linear 3-forests in K_m(n). Therefore, la₃(K_m(n)) ≤ (m − 1) · (²⁽ⁿ⁻²⁾₃ ) + (m − 1) + ^2·₃^m2 = ^2(m−1)n+1₃ = d^2(m−1)n₃ e.

Proposition 3.4.15. la₃(K_m(n)) ≤

l2(m−1)n 3

m

if m ≡ 3 (mod 6) and n ≡ 4 (mod 12).

Proof. By Lemma 3.4.3, K_m(n) has a Kⁿ₂,ⁿ₂-factorization in which there are 2m − 2 Kⁿ₂,ⁿ₂-factors. Hence, from the proof of Proposition 3.2.3, the edges with bipartite differences 2, 3, . . . ,ⁿ₂−1 in those Kⁿ_2,ⁿ₂ of Kⁿ_2,ⁿ₂-factors in K_m(n)can generate (2m−2)

· (²(ⁿ2−2)

3 ) linear 3-forests.

Moreover, the edges with bipartite differences 0, 1 in those Kⁿ_2,ⁿ₂ of Kⁿ_2,ⁿ₂-factors in K_m(n)also can produce 2m−2 linear 3-forests except half of the edges with bipartite difference 1 in those Kⁿ_2,ⁿ₂ of Kⁿ_2,ⁿ₂-factors in K_m(n) which are not being used.

Therefore, in what follows, we want to show that those edges which are not being used also produce linear 3-forests. Since K_2m(ⁿ2) is the union of K^m(n) and one Kⁿ₂,ⁿ₂ -factor of K_2m(ⁿ2), for convenience, we consider this question on K^2m(ⁿ2).

For all i ∈ {0, 1, . . . , 2m − 1}, let the vertices of partite set V_i in K_2m(ⁿ2) be

Then (i) the edges of M₁ and U₂ can produce a linear 3-forest; (ii) the edges of U₃, U₄, . . . , U_m−1 can generate ^2(m−3)₃ linear 3-forests from the proof of Proposition

Proof. It is similar to the proof of Proposition 3.4.15 except the following: (i) The edges of M₁ and M_m can produce a linear 3-forest; (ii) the edges of U₂, U₃, . . . , U_m−1 can generate ^2(m−2)₃ linear 3-forests from the proof of Proposition 3.2.3. Therefore, la₃(K_m(n)) ≤ (2m−2)· Kⁿ_4,ⁿ₄-factors. Hence, from the proof of Proposition 3.2.3, the edges with bipartite differences 2, 3, . . . ,ⁿ₄−1 in those Kⁿ₄,ⁿ₄ of Kⁿ₄,ⁿ₄-factors in K_m(n)can generate (4m−4)

· (²(ⁿ4−2)

3 ) linear 3-forests.

Moreover, the edges with bipartite differences 0, 1 in those Kⁿ_4,ⁿ₄ of Kⁿ_4,ⁿ₄-factors in K_m(n)also can produce 4m−4 linear 3-forests except half of the edges with bipartite difference 1 in those Kⁿ_4,ⁿ₄ of Kⁿ_4,ⁿ₄-factors in K_m(n) which are not being used.

Therefore, in what follows, we want to show that those edges which are not being used also produce linear 3-forests. Since K_4m(ⁿ4) is the union of K^m(n) and three Kⁿ_4,ⁿ₄-factors of K_4m(ⁿ4), for convenience, we consider this question on K^4m(ⁿ4). from the proof of Proposition 3.2.3; (iv) the edges of U_2m−1 can produce a linear 3-forest. Hence, la₃(K_m(n)) ≤ (4m − 4) ·

Proof. It is similar to the proof of Proposition 3.4.17 except the following: (i) The edges of M₁ and M₃ can produce a linear 3-forest; (ii) the edges of M₂ and U₄ can produce a linear 3-forest; (iii) the edges of U₅, U₆, . . . , U_2m−1 and M_2m can generate ^2(2m−4)₃ linear 3-forests from the proof of Proposition 3.2.3. Hence, la₃(K_m(n))

≤ (4m − 4) ·

Proposition 3.4.19. la₃(K_m(n)) ≤

l2(m−1)n 3

m

if m ≡ 0, 8 (mod 12) and n ≡ 1, 5 (mod 6).

Proof. Dividing all m partite sets of K_m(n) into ^m₄ disjoint collections of four partite sets shows that Km(n) is the union of K^m₄(4n) and one K4(n)-factor of Km(n). Since

m

4 ≡ 0, 2 (mod 3) and 4n ≡ 4, 8 (mod 12), from Propositions 3.4.6 and 3.4.12 ∼ 3.4.18, la₃(K_m(n)) ≤ la₃(K_4(n)) + la₃(K^m_4(4n)) ≤ ^2(4−1)n₃ +

»

2(^m4−1)⁽⁴ⁿ⁾

3

¼

=

l2(m−1)n 3

m .

From the propositions given above, we have that la₃(K_m(n)) ≤

l2(m−1)n 3

m

if mn ≡ 0 (mod 4). On the other hand, by Lemma 2.1.5, la₃(K_m(n)) ≥

l2(m−1)n 3

m

if mn ≡ 0 (mod 4). Hence, we determine the linear 3-arboricity of K_m(n) for mn ≡ 0 (mod 4) and conclude the work of this section with the following theorem.

Theorem 3.4.20.

la3(Km(n)) =

»2(m − 1)n 3

¼

when mn ≡ 0 (mod 4).

Concluding Remark. By using the ideas in this section, we can also find la₃(K_m(n)) for quite a few other cases when mn ≡ 2 (mod 4). But, we are not able to finish the whole part at this moment due to several stubborn subcases. As for the cases when mn is odd, they are expected to be more difficult.

We remark finally that the work about the linear 3-arboricity of balanced complete multipartite graphs presented in this section will appear in [29].

Chapter 4

Linear 2-arboricity of Complete

在文檔中 完全多部圖之線性k蔭度 (頁 46-54)