The Linear 4-arboricity of Balanced Complete Bipartite Graphs

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The Linear 4-arboricity of Balanced Complete

Bipartite Graphs

∗

Liancui Zuo

†

_{, Shengjie He, and Ranqun Wang}

Abstract

A lineark-forest is a graph whose components are paths of length at mostk. The lineark-arboricity of a graphG, denoted bylak(G), is the least number of lineark-forests needed to decomposeG. In this paper, it is obtained that

la4(Kn,n) =⌈5n/8⌉forn≡0( mod 5).

Keywords: Linear k-forest; linear k-arboricity; com-plete bipartite graph; perfect matching; bipartite diﬀer-ence

1 Introduction

In this paper, all graphs considered are ﬁnite, undirected, and simple (i. e., loopless and without multiple edges). We refer to [20] for terminology in graph theory. In re-cent years, many parameters and classes of graphs were studied. For example, in [11], diﬀerent properties of the intrinsic order graph were obtained, namely those deal-ing with its edges, chains, shadows, neighbors and de-grees of its vertices, and some relevant subgraphs, as well as the natural isomorphisms between them. In [18], the n-dimensional cube-connected complete graph was studied. In [24, 25], the hamiltonicity, path t-coloring, and the shortest paths of Sierpin´ski-like graphs were re-searched. In [26], the vertex arboricity of integer distance graph G(Dm,k)was obtained.

A decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list. If a graph Ghas a decomposition G1, G2,· · ·, Gd, then we say thatG1, G2,· · · , GddecomposeG, orGcan be decomposed into G1, G2,· · ·, Gd. A linear k−forest is a forest whose components are paths of length at most

k. The linear k−arboricity of a graph G, denoted by

lak(G), is the least number of lineark−forests needed to decompose G. Letxbe a real number, denoted by ⌊x⌋ the maximum integer no more than x, and denoted by ⌈x⌉the minimum integer no less thanx. For any integers

a < b, let[a, b]denote the set of integers{a, a+ 1,· · ·, b} for simplicity. For any positive integer λ, let Pλ be a path onλvertices which has lengthλ−1.

The notion of lineark−arboricity was ﬁrst introduced by Habib and Peroche [13], which is a natural generalization of edge coloring. Clearly, a linear 1-forest is induced by

∗_{Manuscript received August 4,2014. This work was supported}

by NSFC for youth with code 61103073.

†_{Liancui Zuo, Shengjie He and Ranqun Wang are with}

Col-lege of Mathematical Science, Tianjin Normal University, Tianjin, 300387, China. Email: [email protected];[email protected]

a matching, and la1(G) is the edge chromatic number, or chromatic index, χ′₍_G₎ _{of a graph}_G_{. Moreover, the}

lineark−arboricitylak(G)is also a reﬁnement of the or-dinary linear arboricityla(G)(orla∞(G))[14] of a graph

G, which is the case when every component of each for-est is a path with no length constraint. In 1982, Habib and Peroche [12] proposed the following conjecture for an upper bound on lak(G).

Conjecture 1.1. IfGis a graph with maximum degree

∆(G)andk≥2, then

lak(G)≤



  

  

⌈∆(G)·|V(G)| 2⌊k·|kV+1(G)|⌋

⌉, if∆(G) =|V(G)| −1,

⌈∆(G)·|V(G)|+1 2⌊k·|kV+1(G)|⌋

⌉, if∆(G)<|V(G)| −1.

Fork=|V(G)| −1, it is the Akiyama’s conjecture [1].

Conjecture 1.2. [1]la(G)≤ ⌈∆(G₂)+1⌉.

So far, quite a few results on the veriﬁcation of Con-jecture1.1have obtained in the literature, especially for graphs with particular structures, such as trees [5, 6, 13], cubic graphs [4, 16, 19], regular graphs [2, 3], planar graphs [17], balanced complete bipartite graphs [8, 9, 10], balanced complete multipartite graphs [22] and complete graphs [5, 7, 8, 9, 21]. It is obtained that the linear 2 -arboricity, the linear 3-arboricity and the low bound of lineark-arboricity of balanced complete bipartite graph in [8, 9, 10], respectively. In [23], Xue and Zuo ob-tained the linear (n−1)-arboricity of complete multi-partite graph Kn(m). In [15], the linear 6-arboricity of

the graph Km,n was obtained. All the results are co-herent with the corresponding cases of Conjecture 1.1. But for the general graph, this conjecture has not been proved yet.

As for a lower bound on lak(G), it is obvious that the following result holds.

Lemma 1.3. For any graph G with maximum degree

△(G), then

lak(G)≥max{⌈∆(G)

2 ⌉,⌈

|E(G)| ⌊k|V_k₊₁(G)|⌋⌉}.

2 Main results

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Let Kn,n = G(A, B) be a balanced complete bipartite graph with partite sets A and B, whereA={a0, a1,· · · ,

an−1} and B = {b0, b1,· · · , bn−1}. It is deﬁned that

the bipartite dif f erence of an edge apbq in [9] as the value (q−p)( modn). The edge set of Kn,n can be partitioned into n pairwise disjoint perfect matchings

M0, M1,· · ·, Mn−1, whereMj is exactly the set of edges

of bipartite diﬀerence j inKn,n forj∈[0, n−1].

Theorem 2.1. la4(Kn,n) =⌈5n/8⌉forn≡0( mod 5).

Proof. Clearly, by Lemma 1.3,

la4(Kn,n)≥ ⌈n2/⌊4·2n/5⌋⌉=⌈5n/8⌉,

so we need only to prove the upper bound.

It is easy to see that the following Claim1holds.

Claim1.For eacht∈[0,⌊n/8⌋−1], the edges ofM8t+2q∪ M8t+2q+1 other than that in ct,q can form one linear4 -forest, where

ct,q={a2+2t+5(i−1)b2+2t+5(i−1)+8t+2q, a4+2t+5(i−1)b4+2t+5(i−1)+8t+2q+1|i∈[1, n/5]}

forq∈ {0,2},and

ct,q={a3+2t+5(i−1)b3+2t+5(i−1)+8t+2q, a5+2t+5(i−1)b5+2t+5(i−1)+8t+2q+1|i∈[1, n/5]}

forq∈ {1,3}.

For example, ifn= 40, the edges ofM0∪M1other than that in

c0,0={a2+5(i−1)b2+5(i−1),

a4+5(i−1)b4+5(i−1)+1|i∈[1,8]}

can form one linear4-forest (please see Fig. 1). Similarly, the edges ofM2∪M3 other than that in

c0,1={a3+5(i−1)b3+5(i−1)+2,

a5+5(i−1)b5+5(i−1)+3|i∈[1,8]}

can form one linear4-forest. The edges ofM4∪M5other than edges in

c0,2={a2+5(i−1)b2+5(i−1)+4,

a4+5(i−1)b4+5(i−1)+5|i∈[1,8]}

can form one linear4-forest. The edges ofM6∪M7other than edges in

c0,3={a3+5(i−1)b3+5(i−1)+6,

a5+5(i−1)b5+5(i−1)+7|i∈[1,8]}

can form another linear4-forest, and so on.

Claim 2. la4(Kn,n) ≤ 5n/8 for n ≡ 0( mod 8) and

n≡0( mod 5).

By Claim 1, we have obtained n/2 linear 4-forests in total. Thus we have to estimate the number of linear

4-forests induced by the union ofct,q.

It is easy to verify that all edges of ∪3

q=0ct,q can form

n/5pairwise disjoint pathsP5

{b5+2t+5(i−1)+8t+3a5+2t+5(i−1)b5+2t+5(i−1)+8t+7

a5+2t+5(i−1)+7b2+2t+5(i−1)+8t+4+10|i∈[1, n/5]}

andn/5pairwise disjoint 4-cycles

{a3+2t+5(i−1)b3+2t+5(i−1)+8t+2a4+2t+5(i−1)

b4+2t+5(i−1)+8t+5a3+2t+5(i−1)|i∈[1, n/5]},

for eacht∈[0, n/8−1]. Moreover, for eacht∈[0, n/8−

1], we obtain one linear4-forest by deleting edges in

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

q=0ct,q, adding edges of

{aubu+8tl+9|u= 2tl+ 5i+ 1, i∈[1, n/5]}

to the remained edges of∪3

q=0ctl,q, and adding edges of {aubu+8(5l−5)+1|u= 2(5l−5) + 5i−1, i∈[1, n/5]}

to the remained edges of ∪3

q=0c(5l−1),q, where tl∈[5(l− 1),5(l−1) + 3] andl∈[1, n/40].

Hence, la4(Kn,n) ≤ n/2 +n/8 = 5n/8 in the case of

n≡0( mod 40).

For example, if n= 40, fort = 0, the edges of∪3

q=0c0,q

can form8pairwise disjoint paths P5

{b5i+3a5ib5i+7a5i+7b5i+11|i∈[1,8]}

and8 pairwise disjoint4-cycles

{a5i−2b5ia5i−1b5i+4a5i−2|i∈[1,8]}.

(Please see Figure2. Note that the pathsP5 of∪3q=0c0,q

have not been displayed in the ﬁgure.) Moreover, for

t= 0, we obtain one linear4-forest by deleting edges in

{a4+5(i−1)b4+5(i−1)+1|i∈[1,8]}

of4-cycles of∪3q=0c0,q and adding edges of

{a4+2+5(i−1)b4+2+5(i−1)+8+1|i∈[1,8]}

q=0c0,q. Similarly, for t= 1,

we obtain one linear 4-forest by deleting edges in

{a4+2+5(i−1)b4+2+5(i−1)+8+1|i∈[1,8]}

of4-cycles of∪3

q=0c1,q and adding edges of

{a4+2·2+5(i−1)b4+2·2+5(i−1)+8·2+1|i∈[1,8]}

q=0c1,q. Fort= 2, we obtain

one linear4-forest by deleting edges in

{a4+2·2+5(i−1)b4+2·2+5(i−1)+8·2+1|i∈[1,8]}

of4-cycles of∪3

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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 b32 b33 b34 b35 b36 b37 b38 b39 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39

Figure 1. M0∪M1 withc0,0broken and others normal

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 b32 b33 b34 b35 b36 b37 b38 b39 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39

Figure 2. broken edges: the edges deleted from 4-cycles of∪3

q=0c0,q

heavy edges: the edges adding to the remained edges of∪3

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{a4+2·3+5(i−1)b4+2·3+5(i−1)+8·3+1|i∈[1,8]}

of4-cycles of∪3

{a4+2·4+5(i−1)b4+2·4+5(i−1)+8·4+1|i∈[1,8]}

of4-cycles of∪3

{a4+5(i−1)b4+5(i−1)+1|i∈[1,8]}

q=0c4,q.

Hence la4(K40,40)≤25.

Claim 3. la4(Kn,n) ≤ ⌈5n/8⌉ for n ≡ 2( mod 8) and

n≡0( mod 5).

The edges ofMn−2∪Mn−1other than that in

{a4+5(i−1)b2+5(i−1), a1+5(i−1)b5(i−1)|i∈[1, n/5]}

can form one linear 4-forest. Now we have obtained ⌊n/8⌋ ·4 + 1 = n/2 linear 4-forests in total by Claim

1. Next we will estimate the number of linear4-forests induced by the edges that are not used in Kn,n.

It is not diﬃcult to verify that all edges of∪3

q=0ct,q can form n/5pairwise disjoint paths P5

{b5+2t+5(i−1)+8t+3a5+2t+5(i−1)b5+2t+5(i−1)+8t+7

a5+2t+5(i−1)+7b2+2t+5(i−1)+8t+4+10|i∈[1, n/5]}

{a3+2t+5(i−1)b3+2t+5(i−1)+8t+2a4+2t+5(i−1)

b4+2t+5(i−1)+8t+5a3+2t+5(i−1)|i∈[1, n/5]},

for each t ∈ [0,⌊n/8⌋ −1]. Moreover, for each t ∈

[0,⌊n/8⌋ −2], we obtain one linear 4-forest by deleting edges in

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8tl+9|u= 2tl+ 5i+ 1, i∈[1, n/5]}

q=0ctl,q and adding edges of {aubu+8(5l−5)+1|u= 2(5l−5) + 5i−1, i∈[1, n/5]}

q=0c(5l−1),q, wheretl∈[5(l− 1),5(l−1) + 3]andl∈[1,⌊⌊n/8⌋/5⌋]. Fort=⌊n/8⌋ −1, we obtain one linear 4-forest by deleting edges in

{al+5(i−1)bl+5(i−1)+8(⌊n/8⌋−1)+1|i∈[1, n/5]}

of4-cycles of∪3

q=0c(⌊n/8⌋−1),qforl= 4 + 2(⌊n/8⌋ −1). It

is easy to verify that the remained edges, i.e., all edges of

{al+5(i−1)bl+5(i−1)+8(⌊n/8⌋−1)+1, a4+5(i−1)b2+5(i−1),

a1+5(i−1)b5(i−1)|i∈[1, n/5]}

withl= 4 + 2(⌊n/8⌋ −1) can form one linear4-forest.

Hence,la4(Kn,n)≤n/2 + (⌊n/8⌋ −1) + 1 + 1 =n/2 +

⌈n/8⌉=⌈5n/8⌉.

For example, ifn= 10, then the edges ofM0∪M1 other than that in

c0,0={albl, a2+lb2+l+1|l= 5i−3, i∈[1,2]}

can form one linear4-forest. The edges ofM2∪M3other than that in

c0,1={albl+2, a2+lb2+l+3|l= 5i−2, i∈[1,2]}

c0,2={albl+4, a2+lb2+l+5|l= 5i−3, i∈[1,2]}

c0,3={albl+6, a2+lb2+l+7|l= 5i−2, i∈[1,2]}

can form one linear 4-forest. Clearly, the edges of

∪3

q=0c0,q produce two disjoint pathsP5

{bl+3albl+7al+7bl+11|l= 5i, i∈[1,2]}

and two disjoint 4-cycles

{atbt+2a1+tb1+t+5at|t= 5i−2, i∈[1,2]}.

We obtain one linear4-forest by deleting edges

{a4+5(i−1)b4+5(i−1)+1|i∈[1,2]}

of4-cycles of∪3

q=0c0,q. Moreover, the edges ofM8∪M9 other than that in

{a4+5(i−1)b2+5(i−1), a1+5(i−1)b5(i−1)|i∈[1,2]}

can form one linear 4-forest. It is not diﬃcult to verify that the remained edges, i.e., all edges of

{a4+lb4+l+1, a4+lb2+l, a1+lbl|l= 5(i−1), i∈[1,2]}

can form one linear4-forest. Hence,la4(K10,10)≤7.

n≡0( mod 5).

{a6+5(i−1)b2+5(i−1), a3+5(i−1)b5(i−1)|i∈[1, n/5]}

can form one linear 4-forest, and the edges of Mn−2∪

Mn−1 other than that in

{a4+5(i−1)b2+5(i−1), a1+5(i−1)b5(i−1)|i∈[1, n/5]}

can form another one. Now we have obtained ⌊n/8⌋ ·

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{bl+3al−8tbl+7al−8t+7bl+11|l= 10t+ 5i, i∈[1, n/5]}

{albl+8t+2a1+lbl+8t+6al|l= 2t+ 5i−2, i∈[1, n/5]}

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(tl+1)+1|u= 2tl+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−15|u= 2⌊n/8⌋+ 5(i−1), i∈[1, n/5]}

of4-cycles of∪3

q=0c(⌊n/8⌋−2),q and adding edges of

{a1+5(i−1)b5(i−1)|i∈[1, n/5]}

ofMn−1that have not been used. Fort=⌊n/8⌋ −1, we also obtain one linear 4-forest by deleting edges in

{aubu+8⌊n/8⌋−6|u= 2⌊n/8⌋+ 5i−4, i∈[1, n/5]}

of4-cycles of∪3

{a3+5(i−1)b5(i−1)|i∈[1, n/5]}

ofMn−3 that have not been used. It is obvious that the remained edges, i.e., all edges of

{aubu+8(⌊n/8⌋−2)+1, a4+5(i−1)b2+5(i−1),

a6+5(i−1)b2+5(i−1), au−1bu+8(⌊n/8⌋−2)+1|

u= 2⌊n/8⌋+ 5(i−1), i∈[1, n/5]},

can form one linear4-forest.

Hence, la4(Kn,n) ≤ n/2 + (⌊n/8⌋ −2) + 1 + 1 + 1 =

n/2 +⌈n/8⌉=⌈5n/8⌉.

n≡0( mod 5).

{a8+5(i−1)b2+5(i−1), a5+5(i−1)b5(i−1)|i∈[1, n/5]}

can form one linear4-forest, the edges ofMn−4∪Mn−3 other than that in

{a7+5(i−1)b4+5(i−1), a5+5(i−1)b1+5(i−1)|i∈[1, n/5]}

can form one linear 4-forest, and the edges of Mn−2∪

Mn−1 other than edges in

{a4+5(i−1)b2+5(i−1), a1+5(i−1)b5(i−1)|i∈[1, n/5]}

can form another one. Now we have obtained⌊n/8⌋ ·4 + 3 = n/2 linear4-forests in total by Claim 1. Thus we have to estimate the number of linear 4-forests induced by the edges that are not used inKn,n.

It is not diﬃcult to see that all edges of ∪3

{bu+8t+3aubu+8t+7au+7bu+8t+11| u= 2t+ 5i

i∈[1, n/5] }

{aubu+8t+2au+1bu+8t+6au| _iu_∈= 2_[1_{, n/}t+ 5_5]i−2 }

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(tl+1)+1|u= 2tl+ 5i+ 1, i∈[1, n/5]}

q=0ctl,q and adding edges of {aubu+8(5l−5)+1|u= 10l+ 5i−11, i∈[1, n/5]}

{aubu+8⌊n/8⌋−7|u= 2⌊n/8⌋+ 5i−3, i∈[1, n/5]}

of 4-cycles of ∪3

{a5+5(i−1)b5(i−1)|i ∈ [1, n/5]} of Mn−5 that have not been used. For each t ∈ {⌊n/8⌋ −2,⌊n/8⌋ − 3}, we obtain one linear 4-forest by deleting edges in

{aubu+8t+1|u= 2t+ 5i−1, i∈[1, n/5]}

of4-cycles of∪3

q=0ct,qand adding edges of

{aubu+8(t+1)+1|u= 2t+ 5i+ 1, i∈[1, n/5]}

that have been deleted from ∪3

q=0c(t+1),q. It is easy to

verify that the remained edges, i.e., all edges of

{aubu+8(⌊n/8⌋−3)+1, a7+5(i−1)b4+5(i−1),

a8+5(i−1)b2+5(i−1), a5+5(i−1)b1+5(i−1),

a1+5(i−1)b5(i−1), a4+5(i−1)b2+5(i−1)|

u= 2⌊n/8⌋+ 5i−7, i∈[1, n/5]},

can form one linear4-forest.

Hence,la4(Kn,n)≤n/2 + (⌊n/8⌋ −3) + 1 + 1 + 1 + 1 =

n/2 +⌈n/8⌉=⌈5n/8⌉.

n≡0( mod 5).

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linear 4-forests induced by the edges that are not used in Kn,n.

It is obvious that all edges of∪3

q=0ct,qcan formn/5 pair-wise disjoint paths P5

{bu+3au−8tbu+7au−8t+7bu+11| u= 10t+ 5i

i∈[1, n/5] }

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(tl+1)+1|u= 2tl+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−7|u= 2⌊n/8⌋+ 5i−3, i∈[1, n/5]}

of 4-cycles of ∪3

q=0c(⌊n/8⌋−1),q. For each t ∈ {⌊n/8⌋ −

2,⌊n/8⌋ −3}, we obtain one linear 4-forest by deleting

edges in

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(t+1)+1|u= 2t+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−23| _iu_∈= 2_[1_{, n/}⌊n/_5]8⌋+ 5i−7 } ∪Mn−1,

can form another one.

Hence,la4(Kn,n)≤ ⌊n/2⌋+ (⌊n/8⌋ −3) + 1 + 1 + 1 + 1 =

⌈n/2⌉+⌊n/8⌋=⌈5n/8⌉.

n≡0( mod 5).

{a5+5(i−1)b2+5(i−1), a2+5(i−1)b5(i−1)|i∈[1, n/5]}

can form one linear 4-forest. Now we have obtained ⌊n/8⌋ ·4 + 1 =⌊n/2⌋linear4-forests in total by Claim

1. Note that the edges ofMn−1also have not been used. Next we will estimate the number of linear4-forests in-duced by the edges that are not used inKn,n.

{bu+8t+3aubu+8t+7au+7bu+8t+11| u_i_∈= 2_[1_{, n/}t+ 5_5]i }

{aubu+8t+2au+1bu+8t+6au| u= 2t+ 5i−2

i∈[1, n/5] }

{aubu+8t+1|u= 2t+ 5i−1, i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(tl+1)+1|u= 2tl+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−7|u= 2⌊n/8⌋+ 5i−3, i∈[1, n/5]}

of 4-cycles of ∪3

{a2+5(i−1)b5(i−1)|i = 1,2,· · ·, n/5} of Mn−2 that have

not been used. For each t ∈[⌊n/8⌋ −2,⌊n/8⌋ −4], we obtain one linear 4-forest by deleting edges in

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(t+1)+1|u= 2t+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−31, a5ib5i−3|

u= 2⌊n/8⌋+ 5i−9, i∈[1, n/5]} ∪ Mn−1,

can form another linear 4-forest.

Hence, la4(Kn,n)≤ ⌊n/2⌋+ (⌊n/8⌋ −4) + 1 + 3 + 1 =

⌈n/2⌉+⌊n/8⌋=⌈5n/8⌉.

n≡0( mod 5).

{aubu+n−5, au+2bu+n−2|u= 5i−3, i∈[1, n/5]}

{aubu+n−3, au+2bu+n|u= 5i−2, i∈[1, n/5]}

can form one and the edges of Mn−1 can form another one. Now we have obtained ⌊n/8⌋ ·4 + 3 =⌈n/2⌉linear

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the number of linear4-forests induced by the edges that are not used inKn,n.

{bu+3au−8tbu+7au−8t+7bu+11| u_i_∈= 10_[1_{, n/}t+ 5_5]i }

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8tl+9|u= 2tl+ 5i+ 1, i∈[1, n/5]}

to the remained edges of ∪3q=0c(5l−1),q, wheretl∈[5(l− 1),5(l−1) + 3]and l∈[1,⌊n/8⌋/5](note that⌊n/8⌋ ≡

0( mod 5) in this case). It is easy to verify that the remained edges, i.e., all edges of

{aubu+n−5, au+2bu+n−2, au+1bu+n−2, au+3bu+n+1| u= 5i−3, i∈[1, n/5]},

can form another linear4-forest.

Hence,la4(Kn,n)≤ ⌈n/2⌉+⌊n/8⌋+ 1 =⌈5n/8⌉.

n≡0( mod 5).

{a9+5(i−1)b2+5(i−1), a6+5(i−1)b5(i−1)|i∈[1, n/5]}

{a5i+3b5i−2, a5(i+1)b5i+1|i∈[1, n/5]}

can form one, and the edges ofMn−3∪Mn−2other than that in

{a3+5(i−1)b5(i−1), a5+5(i−1)b3+5(i−1)|i∈[1, n/5]}

can form another one. Note that the edges of Mn−1 have not been used. Now we have obtained ⌊n/8⌋ ·4 + 3 = ⌊n/2⌋ linear4-forests in total by Claim 1. In the following we will estimate the number of linear4-forests induced by the edges that are not used in Kn,n.

It is not diﬃcult to obtain that all edges of∪3q=0ct,qcan form n/5pairwise disjoint paths P5

{bu+8t+3aubu+8t+7au+7bu+8t+11| u_i_∈= 2_[1_{, n/}t+ 5_5]i }

{a4+2t+5(i−1)b4+2t+5(i−1)+8t+1|i∈[1, n/5]}

of4-cycles of∪3

{aubu+8(tl+1)+1|u= 2tl+ 5i+ 1, i∈[1, n/5]}

{aubu+8⌊n/8⌋−7|u= 2⌊n/8⌋+ 5i−3, i∈[1, n/5]}

of4-cycles of∪3

q=0c(⌊n/8⌋−1),q, and adding edges of

{a6+5(i−1)b5(i−1)|i∈[1, n/5]}

of Mn−6 that have not been used. It is obvious that all edges of

{aubu+8⌊n/8⌋−7|

u= 2⌊n/8⌋+ 5i−3

i∈[1, n/5] } ∪Mn−1

can form one linear4-forest, and the remained edges, i.e., all edges of

{a5i+4b5i−3, a5i+3b5i−2, a5(i+1)b5i+1,

a5i−2b5(i−1), a5ib5i−2|i∈[1, n/5]},

can form another one.

Hence, la4(Kn,n)≤ ⌊n/2⌋+ (⌊n/8⌋ −1) + 1 + 1 + 1 =

⌈n/2⌉+⌈n/8⌉=⌈5n/8⌉.

Combining Claims 2 −9, the theorem is proved com-pletely.

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