The Linear 5-Arboricity
of
Complete Multipartite Graphs
Student: 林和傑 同學 (Ho-Chieh Lin)
Advisor: 嚴志弘 博士 (Chih-Hung Yen)
Contents

Introduction

Known Results

Our Findings

Conclusion

Future Works
Introduction
All graphs considered here are
finite, simple, and undirected.
Definition

A decomposition of a graph G is a list of subgraphs of
G such that each edge of graph G appears in exactly
one subgraph in the list.

Example:
1
G1
K4
G2
2
H1
H2
H3
Definition

A linear k-forest is a graph whose components are
paths of length at most k.

The linear k-arboricity of a graph G, denoted lak(G) ,
is the least number of linear k-forests needed to
decompose G.

Example: la5(K6)=?
K6
la5(K6)=3
G1
G2
G3
Definition
In terms of the “linear k-arboricity problem,” much
attention has been focused on its two extremities:

First, when k is infinite, la∞ (G) that represents the
case when paths of each component have unlimited
lengths is the ordinary linear arboricity of G, i.e. la(G).

Second, when k is 1, la1(G) is the edge chromatic
number, or chromatic index of G, i.e. χ'(G). Then
certain problem is equivalent to the “edge coloring
problem.”
Definition

A complete multipartite graph G is an m-partite
graph and m ≥ 2 such that the edge uv in E(G) if
and only if u and v are in different partite sets.

We write Kn1,
n2, …, nm,
for the complete multipartite
A
complete
graph with partite sets of sizes: n1, n2, …, nm.

balanced
multipartite
graph
is
a
complete multipartite graph with partite sets of the
same size.

We
write
Km(n),
for
the
balanced
complete
multipartite graph with m partite sets of the same
size n.
Known Results on lak(G)
Known Results – Conjecture

Conjecture. (1982, Habib & Peroche)
If G is a graph with maximum degree ∆(G) and k ≥ 2,
then
   (G ) | V (G ) | 

 if  (G )  | V (G ) | 1 and
k |V ( G )|
  2  k 1  

la k (G )  

   (G ) | V (G ) | 1 if  (G )  | V (G ) | 1.

k |V ( G )|
 
2
 k 1  

Known Results - Cubic graph
Year
Scholars
1984 Bermond et al.
1996
Jackson &
Wormald
Results
If G is a cubic graph with la3(G)=2, then
|V(G)|≡0 (mod 4).
If G is a cubic graph and k ≥ 18,
then lak(G) = 2.
If G is a cubic graph and k ≥ 5,
1999
Thomassen
then lak(G) ≤ 2.
Known Results – Other Regular graph
Year
Scholars
Results
There is an absolute constant c  0
such that for every d - regular graph G
2001
Alon et al.
and every d  k  2,
(k  1)d
la k (G ) 
 c kd log d .
2k
Known Results - Tree
Year
Scholars
Results
1. If T is a tree with ∆(T)=2θ－1, then
1999
Chang
lak(T)=θ for k ≥ 2.
2. If T is a tree with ∆(T)=2θ,
then θ≤ lak(T) ≤θ＋1 for k ≥ 2.
Known Results – Complete graph
Year
Scholars
Results
1984 Bermond et al. For m≠10, 11 (mod 12), la2(Km)=
 m(m  1) 

.
2m
 2 3  
 3(3u  1) 
 3(3u  1) 
la 2 ( K 3u )  
,
la
(
K
)

2
3u 1


4 
4



1991
Chen et al.
2008
Fu et al.
To
appear
Yen & Fu
 (3u  2)(3u  1) 
and la 2 ( K 3u  2 )  

 2(2u  1) 
except possibly if 3u  1  49, 52, 58.
 m(m  1) 
la3 ( K m )  
.
3m
 2 4  
la2 ( K12t 10 )  la2 ( K12t 11 )  9t  8 for any t.
Known Results – Complete graph
Year
2002
2007
Scholars
Chen &
Huang
Lee & Lin
Results
m
 m 
Suppose m  i  2 and let    1  k  
 2.

i
 i  1
 m(m  1) 
Then la k ( K m )  
,

 2(m  i ) 
and the equality holds in case that i  2.
m
m
If    1  k     2,
3
2
 m(m  1)   m 
then la k ( K m )  
    2.

 2(m  3)   2 
Known Results – Complete Bipartite Graph
Year
Scholars
Results
1994
 n2 
Fu & Huang la2(Kn,n) = 
.
4n 
  3 
s
2005
Yen & Fu
1. If s  2r , then la 2(K r,s ) 
 2  .
2. If 5  t  0 and r  t  1,
then la 2 ( K r , 2 r t )  r.
2008
Fu et al.
 n2 
la3 ( K n,n )   3n 
  2 
Known Results – Complete Bipartite Graph
Year
2002
Scholars
Chen &
Huang
Results
 2n 
 2n 
Suppose 2n  i  2 and let    1  k  
 2.

 i 
 i  1
 n2 
Then la k ( K n ,n )  
,
 2n  i 
and the equality holds in case that i  2.
Known Results – Km(n)
Year
Scholars
Results
 3(m  1)n 
la 2 ( K m ( n ) )  
 if m and n satisfy
4


one of the following properties :
2005
Yen & Fu
(1) m  3 (mod 12) and n  1 (mod 2);
(2) m  0 (mod 4) and n  0 (mod 6);
(3) m  2 (mod 4) and n  0 (mod 3);
(4) m  0 (mod 3) and n  0 (mod 2).
2007
Yen & Fu
 2(m  1)n 
la3 ( K m ( n ) )  
when mn  0 (mod 4).

3


Known Results – Planar Graph
Year
Scholars
Results
Let G be a planar graph with maximum degree 
and girth g . Then,
2003
Lih et al.
   1
 12;
(1) la 2 (G )  

 2 
   1
 6 if g  4;
(2) la 2 (G )  

 2 
   1
 2 if g  5;
(3) la 2 (G )  

 2 
   1
 1 if g  7.
(4) la 2 (G )  

 2 
Known Results
lak(G)
Types of Graph G
Cubic Graph
Tree
Complete Graph
Km
Complete Bipartite Graph
Km,n
Balanced Complete Bipartite Graph
Kn,n
Balanced Complete Multipartite Graph
Km(n)
k=2
k=3
k=5
Our Findings on la5(Kn,n)
Our Findings on la5(Kn,n)-Steps

Step 1: Calculate lower bounds of la5(Kn,n).

Step 2: Propose decomposition methods to
obtain upper bounds of la5(Kn,n).

Step 3: Determine the exact values of la5(Kn,n).
Properties of Linear k-Arboricity

Lemma.
Δ(G ) 

lak (G) ≥ max {
 2 ,


 E (G )
 k  V (G )
 k 1


 }.


Lower Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
15t
9t
15t+8
9t+5
15t+1
9t+1
15t+9
9t+6
15t+2
9t+2
15t+10
9t+7
15t+3
9t+2
15t+11
9t+7
15t+4
9t+3
15t+12
9t+8
15t+5
9t+4
15t+13
9t+9
15t+6
9t+4
15t+14
9t+9
15t+7
9t+5
≤
Base Concept

Let G(X,Y) be a bipartite graph with partite sets
X={x0, x1,…, xr-1} and Y={y0, y1,…, ys-1}. Suppose
that |Y| = s ≥ r = |X|. We define the bipartite
difference (B.D.) of an edge xpyq in G(X,Y) as the
value

Ex:
q - p (mod s).
x0
x1
K2,3
y0
y1
y2
Base Concept

In Kn,n, a set consisting of those edges with the
same bipartite difference is a perfect matching.

Note that edges of Kn,n have n distinct bipartite
differences.

Example:
x0
x1
x2
y0
y1
y2
K3,3
Our Findings on la5(Kn,n)

Lemma 1.
If n ≥ 6 is a multiple of 3 and α in { 0, 1,…, n-5 },
then the edges of bipartite difference α, α+1,
α+2, α+3, α+4 in Kn,n can form three pairwise
edge-disjoint linear 5-forests.

Example.
In K15,15, all edges of bipartite difference 0, 1, 2, 3, 4
can form three pairwise edge-disjoint linear 5-forests.
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
Figure 1.
B.D.= 0, 1
Example:
(Red dashed
lines are unused
edges of B.D. 0)
In K15,15, three
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
full linear 5-
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
forests
Figure 2.
formed by
B.D.= 2, 3
(Blue dashed
lines are
unused edges
of B.D. 3)
edges of
Bipartite
Difference
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
(B.D.) 0, 1, 2,
Figure 3.
3, 4
B.D.= 4
with previously
unused edges
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
15t
9t
15t+8
9t+5
15t+1
9t+1
15t+9
9t+6
15t+2
9t+2
15t+10
9t+7
15t+3
9t+2
15t+11
9t+7
15t+4
9t+3
15t+12
9t+8
15t+5
9t+4
15t+13
9t+9
15t+6
9t+4
15t+14
9t+9
15t+7
9t+5
≤
Our Method for n ≡ 0 (mod 15) :
Take K15,15 for example.

We group edges of Bipartite Difference (B. D.)
0, 1, 2, 3, 4 as the 1st group;
Form 3 linear 5-forests
5, 6, 7, 8, 9 as the 2nd group;
Form 3 linear 5-forests
10, 11, 12, 13, 14 as the 3rd group.
Form 3 linear 5-forests
Thus 9 linear 5-forests are needed totally to decompose K15,15
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
15t+1
9t
n=
la5(Kn, n) la5(Kn, n)
≥
15t+8
9t+5
9t+1
15t+9
9t+6
15t+2
9t+2
15t+10
9t+7
15t+3
9t+2
15t+11
9t+7
15t+4
9t+3
15t+12
9t+8
15t+5
9t+4
15t+13
9t+9
15t+6
9t+4
15t+14
9t+9
15t+7
9t+5
9t+2
9t+4
≤
9t+6
9t+8
Properties of Linear k-Arboricity

Lemma.
If H is a subgraph of G,
then lak(H) ≤ lak(G).
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
15t+1
9t+1
15t+2
9t+2
15t+3
9t+2
15t+4
9t+3
15t+5
9t+4
15t+6
9t+4
15t+7
9t+5
9t
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t+8
9t+5
15t+9
9t+6
9t+2
15t+10
9t+7
9t+2
15t+11
9t+7
15t+12
9t+8
9t+8
9t+4
15t+13
9t+9
9t+9
9t+4
15t+14
9t+9
9t+9
15t+15
9t+9
9t+9
9t+6
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
9t
15t+8
9t+5
?
15t+1
9t+1
?
15t+9
9t+6
9t+6
15t+2
9t+2
9t+2
15t+10
9t+7
?
15t+3
9t+2
9t+2
15t+11
9t+7
?
15t+4
9t+3
?
15t+12
9t+8
9t+8
15t+5
9t+4
9t+4
15t+13
9t+9
9t+9
15t+6
9t+4
9t+4
15t+14
9t+9
9t+9
15t+7
9t+5
?
Our Strategy for n ≡ 8 (mod 15) :
Take K23,23 for example.

We consider edges of Bipartite Difference (B. D.)
0, 1, 2, 3, 4 as the 1st group;
Form 3 linear 5-forests + 1 green edge
5, 6, 7, 8, 9 as the 2nd group;
Form 3 linear 5-forests + 1 green edge
10, 11, 12, 13, 14 as the 3rd group;
Form 3 linear 5-forests + 1 green
15, 16, 17, 18, 19 as the 4th group;
Form 3 linear 5-forests + 1 green
20, 21 as the 2nd to last group;
and 22 as the last group.
Form 1 linear 5-forests + pink edges
Form 1 linear 5-forest
The Last Linear 5-Forest of K23,23
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
x20
x21
x22
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
y21
y22
In K23,23 , an unfull linear 5-forest formed by edges of B. D. 22 and
all previously unused green edges and previously unused pink edges
Our Strategy for n ≡ 8 (mod 15) :
Take K23,23 for example.

We group edges of Bipartite Difference (B. D.)
0, 1, 2, 3, 4 as the 1st group;
Form 3 linear 5-forests
5, 6, 7, 8, 9 as the 2nd group;
Form 3 linear 5-forests
10, 11, 12, 13, 14 as the 3rd group;
Form 3 linear 5-forests
15, 16, 17, 18, 19 as the 4th group;
Form 3 linear 5-forests
20, 21 as the 2nd to last group;
and 22 as the last group.
Form 1 linear 5-forest
Form 1 linear 5-forest
Thus 14 linear 5-forests are needed totally to decompose K23,23
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
9t
15t+8
9t+5
9t+5
15t+1
9t+1
?
15t+9
9t+6
9t+6
15t+2
9t+2
9t+2
15t+10
9t+7
?
15t+3
9t+2
9t+2
15t+11
9t+7
?
15t+4
9t+3
?
15t+12
9t+8
9t+8
15t+5
9t+4
9t+4
15t+13
9t+9
9t+9
15t+6
9t+4
9t+4
15t+14
9t+9
9t+9
15t+7
9t+5
?
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
9t
15t+8
9t+5
9t+5
15t+1
9t+1
9t+1
15t+9
9t+6
9t+6
15t+2
9t+2
9t+2
15t+10
9t+7
?
15t+3
9t+2
9t+2
15t+11
9t+7
9t+7
15t+4
9t+3
9t+3
15t+12
9t+8
9t+8
15t+5
9t+4
9t+4
15t+13
9t+9
9t+9
15t+6
9t+4
9t+4
15t+14
9t+9
9t+9
15t+7
9t+5
?
Lower and Upper Bounds of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
9t
15t+8
9t+5
9t+5
15t+1
9t+1
9t+1
15t+9
9t+6
9t+6
15t+2
9t+2
9t+2
15t+10
9t+7
9t+7
15t+3
9t+2
9t+2
15t+11
9t+7
9t+7
15t+4
9t+3
9t+3
15t+12
9t+8
9t+8
15t+5
9t+4
9t+4
15t+13
9t+9
9t+9
15t+6
9t+4
9t+4
15t+14
9t+9
9t+9
15t+7
9t+5
9t+5
Exact Values of la5(Kn,n)
n=
la5(Kn, n) la5(Kn, n)
≥
≤
n=
la5(Kn, n) la5(Kn, n)
≥
≤
15t
9t
9t
15t+8
9t+5
9t+5
15t+1
9t+1
9t+1
15t+9
9t+6
9t+6
15t+2
9t+2
9t+2
15t+10
9t+7
9t+7
15t+3
9t+2
9t+2
15t+11
9t+7
9t+7
15t+4
9t+3
9t+3
15t+12
9t+8
9t+8
15t+5
9t+4
9t+4
15t+13
9t+9
9t+9
15t+6
9t+4
9t+4
15t+14
9t+9
9t+9
15t+7
9t+5
9t+5
Our Findings

Theorem.
 n2 
la5(Kn,n) = 
for all n.
5n 
  3 
Our Findings on la5(K3(n))
Our Findings on la5(K3(n))-Steps

Step 1: Calculate lower bounds of la5(K3(n)).

Step 2: Propose decomposition methods to
obtain upper bounds of la5(K3(n)).

Step 3: Determine the exact values of la5(K3(n)).
Properties of Linear k-Arboricity

Lemma.
Δ(G ) 

lak (G) ≥ max {
 2 ,


 E (G )
 k  V (G )
 k 1


 }.


Lower Bounds of la5(K3(n))
n=
la5(K3(n)) la5(K3(n))
≥
≤
n=
la5(K3(n)) la5(K3(n))
≥
≤
10t
12t
10t+5
12t+7
10t+1
12t+2
10t+6
12t+8
10t+2
12t+3
10t+7
12t+9
10t+3
12t+4
10t+8
12t+10
10t+4
12t+5
10t+9
12t+12
Base Concept
x0
K3(2)
x0
x1
y0
y1
z0
z1
x0
x1
x1
y0
y1
z0
z1
Our Findings on la5(K3(n))

Lemma 1.
If n ≥ 6 is even and α in { 0, 2, 4,…, n-6 },
then all edges of bipartite difference α, α+1, α+2,
α+3, α+4 in K3(n) can form six pairwise edgedisjoint linear 5-forests.

Example.
In K3(10), all edges of bipartite difference 0, 1, 2, 3, 4
can form six pairwise edge-disjoint linear 5-forests.
Type I: The 5 B.D. are starting from an even number.
Take edges of B.D. 0, 1, 2, 3, 4 in K3(10) for example.
x0
X
0
Y
x1
x2
x3
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 0, 1
1
y0
y1
y2
y3
2
Z
x4
z0
y4
y5
y6
y7
y8
y9
z4
z5
z6
z7
z8
z9
4
z1
z2
z3
Z, X –
edges of
B.D. 3
3
X
x0
x1
x2
x3
Y, Z –
edges of
B.D. 2, 4
x4
x5
x6
A path of length 5 in K3(10)
x7
x8
x9
The 1st Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 0, 1
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 2, 4
Z, X –
edges of
B.D. 3
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 2nd Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 0, 1
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 2, 4
Z, X –
edges of
B.D. 3
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 1st Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 0, 1
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 2, 4
Z, X –
edges of
B.D. 3
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 3rd Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 3
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 0, 1
Z, X –
edges of
B.D. 2, 4
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 4th Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 3
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 0, 1
Z, X –
edges of
B.D. 2, 4
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 5th Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 2, 4
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 3
Z, X –
edges of
B.D. 0, 1
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
The 6th Linear 5-Forest of K3(10)
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
X, Y –
edges of
B.D. 2, 4
Y
Z
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
z0
z1
z2
z3
z4
z5
z6
z7
z8
z9
Y, Z –
edges of
B.D. 3
Z, X –
edges of
B.D. 0, 1
X
x0
x1
x2
x3
x4
x5
x6
x7
x8
A linear 5-forest (full)
formed by partial edges of B.D. 0, 1, 2, 3, 4 in K3(10)
x9
Our Findings on la5(K3(n))

Lemma 2.
If n ≥ 6 is even and α in { 1, 3, 5,…, n-5 },
then all edges of bipartite difference α, α+1, α+2,
α+3, α+4 in K3(n) can form six pairwise edgedisjoint linear 5-forests.

Example.
In K3(10), all edges of bipartite difference 5, 6, 7, 8, 9
can form six pairwise edge-disjoint linear 5-forests.
Our Findings on la5(K3(n))

Lemma 1.
If n ≥ 6 is even and α in { 0, 2, 4,…, n-6 }, then all
edges of bipartite difference α, α+1, α+2, α+3, α+4 in K3(n)
can form six pairwise edge-disjoint linear 5-forests.

Lemma 2.
If n ≥ 6 is even and α in { 1, 3, 5,…, n-5 }, then all
edges of bipartite difference α, α+1, α+2, α+3, α+4 in K3(n)
can form six pairwise edge-disjoint linear 5-forests.
Lower and Upper Bounds of la5(K3(n))
n=
la5(K3(n)) la5(K3(n))
≥
≤
10t
12t
10t+1
12t+2
10t+2
12t+3
10t+3
12t+4
10t+4
12t+5
12t
12t+3
12t+5
n=
la5(K3(n)) la5(K3(n))
≥
≤
10t+5
12t+7
10t+6
12t+8
10t+7
12t+9
10t+8
12t+10
12t+10
10t+9
12t+12
12t+12
12t+8
Exact Values of la5(K3(n))
n=
la5(K3(n)) la5(K3(n))
≥
≤
n=
la5(K3(n)) la5(K3(n))
≥
≤
10t
12t
12t
10t+5
12t+7
?
10t+1
12t+2
?
10t+6
12t+8
12t+8
10t+2
12t+3
12t+3
10t+7
12t+9
?
10t+3
12t+4
?
10t+8
12t+10
12t+10
10t+4
12t+5
12t+5
10t+9
12t+12
12t+12
Our Findings

Theorem.
 3n 2 
la5(K3(n)) =  5n 
  2 
when n ≡ 0, 2, 4, 6 ,8 ,9 (mod 10)
Our Findings on la5(Km(n))
Our Findings

la5(K6(n)) = 3n.

la5(Km(n)) =
3(m  1)n
5
if m and n satisfy one of the
following properties:
(a) n ≡ 0 (mod 30).
(b) m ≡ 0 (mod 2) and n ≡ 0 (mod 15).
(c) m ≡ 0 (mod 3) and n ≡ 0 (mod 10).
(d) m ≡ 0 (mod 6) and n ≡ 0 (mod 5).
Conclusion

Within
the
limitations
of
the
study,
the
following
conclusions are warranted:
1.
la5(Kn,n) is determined.
2.
la5(K3(n)) is determined if n≡0, 2, 4, 6 ,8 ,9 (mod 10).
3.
la5(K6(n)) = 3n.
4.
la5(Km(n)) is determined
if m and n satisfy one of the following properties:
(a) n ≡ 0 (mod 30).
(b) m ≡ 0 (mod 2) and n ≡ 0 (mod 15).
(c) m ≡ 0 (mod 3) and n ≡ 0 (mod 10).
(d) m ≡ 0 (mod 6) and n ≡ 0 (mod 5).
Future works

The following problems are major goals of our
future study.
1.
la5(K3(n)), when n ≡ 1, 3, 5, 7 (mod 10).
2.
la5(Km).
3.
la4(Kn,n).
4.
There also seems to be a connection between
the value of m, n and the strategies we adopt
to decompose a Km(n).
The End
Thank you
for your attention
Our Strategy for n ≡ 8 (mod 15) :
Take K23,23 for example.

We consider edges of Bipartite Difference (B. D.)
0, 1, 2, 3, 4 as the 1st group;
Form 3 linear 5-forests + 1 green edge
5, 6, 7, 8, 9 as the 2nd group;
Form 3 linear 5-forests + 1 green edge
10, 11, 12, 13, 14 as the 3rd group;
Form 3 linear 5-forests + 1 green
15, 16, 17, 18, 19 as the 4th group;
Form 3 linear 5-forests + 1 green
20, 21 as the 2nd to last group;
and 22 as the last group.
The Way We Deal with Green Edges
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
x20
x21
x22
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
y21
y22
In K23,23 , an unfull linear 5-forest formed by edges of B. D. = 22 and
previously unused green edges of all groups
Our Strategy for n ≡ 8 (mod 15) :
Take K23,23 for example.

We consider edges of Bipartite Difference (B. D.)
0, 1, 2, 3, 4 as the 1st group;
Form 3 linear 5-forests + 1 green edge
5, 6, 7, 8, 9 as the 2nd group;
Form 3 linear 5-forests + 1 green edge
10, 11, 12, 13, 14 as the 3rd group;
Form 3 linear 5-forests + 1 green
15, 16, 17, 18, 19 as the 4th group;
Form 3 linear 5-forests + 1 green
20, 21 as the 2nd to last group;
and 22 as the last group.
Form 1 linear 5-forests + pink edges
The 2nd to Last Linear 5-Forest of
K23,23
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
x20
x21
x22
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
y21
y22
In K23,23 , a full linear 5-forest formed by edges of B. D. = 20, 21
The Difficulty We Met
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
x20
x21
x22
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
y21
y22
In K23,23 , an unfull linear 5-forest formed by edges of B. D. = 22 and
all previously unused green edges and previously unused pink edges
The 2nd to Last Linear 5-Forest of
K23,23
x0
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
x15
x16
x17
x18
x19
x20
x21
x22
y0
y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
y21
y22
In K23,23 , an unfull linear 5-forest formed by edges of B. D. = 20, 21