L(p, q)-labelings of
subdivision of graphs
Meng-Hsuan Tsai
蔡孟璇
指導教授：郭大衛教授
國立東華大學
應用數學系碩士班
Outline:
• Introduction
• Main result
• 𝜆 𝐺
3
= Δ + 1 with Δ 𝐺 ≥ 4
• Δ 𝐺 ≥ 5, ℎ(𝑒) ≥ 3 ⟹ 𝜆 𝐺
ℎ
=Δ+1
• Δ 𝐺 ≥ 4, ℎ(𝑒) ≥ 4 ⟹ 𝜆 𝐺
ℎ
=Δ+1
• Reference
Introduction
Definition:
• L(p,q)-labeling of G：
𝑑𝐺 𝑢, 𝑣 = 1 ⇒ 𝑓 𝑢 − 𝑓(𝑣) ≥ 𝑝
∀𝑢, 𝑣 ∈ 𝑉(𝐺)
𝑑𝐺 𝑢, 𝑣 = 2 ⇒ 𝑓 𝑢 − 𝑓(𝑣) ≥ 𝑞
• A k-L(p,q)-labeling is an L(p,q)-labeling such that no
label is greater than k.
• The L(p,q)-labeling number of G, denoted by 𝜆𝑝,𝑞 (𝐺),
is the smallest number k such that G has a k-L(p,q)labeling.
• 𝜆 𝐺 = 𝜆2,1 (𝐺).
Example:
• L(2,1)-labeling of 𝐶4 ：
0
2
0
3
6
4
4
1
6-L(2,1)-labeling
4-L(2,1)-labeling
𝜆 𝐺 =4
Griggs and Yeh (1992)
• They showed that the L(2,1)-labeling problem
is NP-complete for general graphs and proved
that 𝜆 𝐺 ≤ ∆2 𝐺 + 2∆(𝐺).
• They conjectured that 𝜆 𝐺 ≤ Δ 𝐺 for any
graph G with ∆(𝐺) ≥ 2.
Gonçalves (2005)
• 𝜆 𝐺 ≤ ∆2 𝐺 + ∆ 𝐺 − 2 with ∆(𝐺) ≥ 2.
Georges and Mauro (1995)
• They studied the L(2,1)-labeling of the incident
graph G, which is the graph obtained from G by
replacing each edge by a path of length 2.
• Example：
C3
Definition:
• Given a graph G and a function ℎ from 𝐸(𝐺) to ℕ, the
ℎ − 𝑠𝑢𝑏𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 of G, denoted by 𝐺(ℎ) , is the graph
obtained from G by replacing each edge 𝑢𝑣 in G with a
1
2
𝑛−1
path 𝑃: 𝑢(𝑥𝑢𝑣
)ℎ (𝑥𝑢𝑣
)ℎ … (𝑥𝑢𝑣
)ℎ 𝑣, where 𝑛 = ℎ(𝑢𝑣).
• If ℎ 𝑒 = 𝑐 for all 𝑒 ∈ 𝐸(𝐺), we use 𝐺(𝑐) to replace 𝐺(ℎ) .
• Example :
𝐺 = 𝐶4
𝐺(3)
𝐺(5)
Lü
• Lü studied the 𝐿 2,1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 of 𝐺(𝑐) and
conjectured that 𝜆 𝐺 3 ≤ ∆ + 2 for and graph with
maximum degree ∆.
Karst
• Karst et al. studied the 𝐿 𝑑, 1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 of
𝐺(𝑐) and gave upper bound for 𝜆 𝐺 𝑐 (where 𝑐 ≥ 3).
• Karst et al. showed that 𝜆𝑑 𝐺
𝑚𝑎𝑥
∆ 𝐺
2
3
≤𝑑+
∆ 𝐺
2
+
, 𝑑 − 1 for any graph G with ∆(𝐺) ≥ 3.
 From this, we have 𝜆 𝐺 3 ≤ ∆ + 2 for all graph G with
∆≥ 3 and 𝜆 𝐺 = Δ + 1 with Δ(𝐺) is even and Δ(𝐺) ≠ 2.
Extend
𝐿 𝑝, 𝑞 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 of subdivisions of graphs.
• 𝜆 𝐺
3
= Δ + 1 for any graph G with Δ(𝐺) ≥ 4.
• 𝜆 𝐺
ℎ
=∆+1
if ∆(𝐺) ≥ 5 and ℎ is a function from 𝐸(𝐺) to ℕ so
that ℎ 𝑒 ≥ 3 for all 𝑒 ∈ 𝐸(𝐺),
or if ∆(𝐺) ≥ 4 and ℎ is a function from 𝐸(𝐺) to ℕ so
that ℎ 𝑒 ≥ 4 for all 𝑒 ∈ 𝐸(𝐺).
𝐿 𝑝, 𝑞 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟 of 𝐺(3)
for ∆(𝐺) ≤ 2
Georges and Mauro
• They studied the L(p,q)-labeling numbers of
paths and cycles and gave the following results.
Theorem 1:
• For all 𝑛 ≥ 1,
𝜆𝑝,𝑞 𝑃𝑛
𝑖𝑓 𝑛 = 1,
0,
𝑖𝑓 𝑛 = 2,
𝑝,
𝑖𝑓 𝑛 = 3 𝑜𝑟 4,
= 𝑝 + 𝑞,
𝑝 + 2𝑞,𝑖𝑓 𝑛 ≥ 5 𝑎𝑛𝑑 𝑝 ≥ 2𝑞,
2𝑝, 𝑖𝑓 𝑛 ≥ 5 𝑎𝑛𝑑 𝑝 ≤ 2𝑞.
Theorem 2:
• For all 𝑛 ≥ 3, if 𝑝 ≥ 2𝑞, then
𝜆𝑝,𝑞 𝐶𝑛
𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑,
2𝑝,
𝑖𝑓 𝑛 ≡ 0 𝑚𝑜𝑑 4 ,
𝑝 + 2𝑞,
=
2𝑝, 𝑖𝑓 𝑛 ≡ 2 𝑚𝑜𝑑 4 𝑎𝑛𝑑 𝑝 ≤ 3𝑞,
𝑝 + 3𝑞, 𝑖𝑓 𝑛 ≡ 2 𝑚𝑜𝑑 4 𝑎𝑛𝑑 𝑝 ≥ 3𝑞.
• If 𝑝 ≤ 2𝑞, then
𝜆𝑝,𝑞 𝐶𝑛
2𝑝, 𝑖𝑓 𝑛 ≡ 0 𝑚𝑜𝑑 3 ,
4𝑞,
=
𝑖𝑓 𝑛 = 5,
𝑝 + 2𝑞, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Theorem 3:
• If ∆(𝐺) ≤ 2, then
𝜆𝑝,𝑞 𝐺
3
𝑝 + 𝑞,
𝑖𝑓 Δ 𝐺 = 1
𝑝 + 2𝑞, 𝑖𝑓 𝑝 ≥ 2𝑞 𝑎𝑛𝑑 𝐺 ∈ 𝒜
=
𝑝 + 3𝑞, 𝑖𝑓 𝑝 ≥ 3𝑞 𝑎𝑛𝑑 𝐺 ∈ ℬ
2𝑝,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
where
𝒜 = 𝐺: 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑜𝑓 𝑒𝑣𝑒𝑟𝑦 𝑐𝑦𝑐𝑙𝑒 𝑖𝑛 𝐺 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4
ℬ = 𝐺: 𝐺 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑎 𝑐𝑦𝑐𝑙𝑒 𝐶𝑛 , 𝑛 ≡ 2 (𝑚𝑜𝑑 4) .
𝐿 2,1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟 of 𝐺(3)
Definition:
• An 𝐿 𝑝, 𝑞 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓of 𝐺(ℎ) is said to be 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 if
𝑓 𝑢 = 0 for all 𝑢 ∈ 𝑉(𝐺).
• The 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟 of 𝑣 in G, denoted 𝑁𝐺 (𝑣), is defined by
𝑁𝐺 𝑣 = 𝑢: 𝑢𝑣 ∈ 𝐸(𝐺) .
• If 𝑓 is a 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑘 − 𝐿 2,1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 of 𝐺(3) and 𝑣 ∈
𝑉 𝐺 , then we use 𝐴𝑓 (𝑣) to denoted the set 2,3,4, … 𝑘 −
𝑓 𝑢 : 𝑢 ∈ 𝑁𝐺 3 (𝑣) .
• An 𝐿 2,1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐺(ℎ) is said to be a
𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 of 𝐺(ℎ) if 𝑓 is 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 and
𝜆𝑓 𝐺
ℎ
= ∆ + 1.
Definition:
• Let 𝑓 be an 𝐿 𝑝, 𝑞 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 of G. For 𝑢 ∈ 𝑉(𝐺), a
trail 𝑢 = 𝑢1 𝑢2 … 𝑢𝑙 in 𝐺(3) is called 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑛𝑔 −
𝑎, 𝑐; 𝑏, 𝑑 − 𝑡𝑟𝑎𝑖𝑙 in 𝐺(3) corresponding to 𝑓 if
•
𝑓
𝑓
𝑓
𝑓
𝑓
𝑢𝑖
𝑢𝑖
𝑢𝑖
𝑢𝑖
𝑢𝑖
= 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≡ 1 (𝑚𝑜𝑑 3)
= 𝑎 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≡ 2 (𝑚𝑜𝑑 6)
= 𝑐 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≡ 3 (𝑚𝑜𝑑 6)
= 𝑏 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≡ 5 (𝑚𝑜𝑑 6)
= 𝑑 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≡ 0 (𝑚𝑜𝑑 6)
0
0
a c
0
bd
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
pf：
𝑢
3
= Δ + 1.
𝑣
𝐺∗
𝐸(𝐺 ∗ ) is minimum
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
pf：
𝑢
𝑣
𝐻
3
= Δ + 1.
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 1: For any 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , 2 ∉
𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣) and ∆ + 1 ∉ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣).
Lemma 1:
𝑊: 𝑢 = 𝑢1 𝑢2 … 𝑢𝑙 𝑖𝑠 𝑎 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑛𝑔 −
∆ − 1, ∆ + 1; ∆ − 1, ∆ + 1 − 𝑡𝑟𝑎𝑖𝑙
𝑢
𝑢
Δ+1
Δ−1
Δ−1
Δ+1
Δ+1
Δ−1
Δ−1
Δ+1
Δ+1
Δ+1
Δ−1
Δ−1
Δ+1
Δ−1
Δ+1
Δ+1
Δ−1
Δ−1
Δ+1
𝑢𝑙
Δ−1
Δ−1
a
∵ Δ + 1 ∈ 𝐴𝑓 (𝑢𝑙 )
Δ+1
𝑢𝑙
∵ 𝑎必定 ≤ ∆ − 3
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 1: For any 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , 2 ∉
𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣) and ∆ + 1 ∉ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣).
• Suppose ∆ + 1 ∈ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣)
𝑣
∆+1 ∆+1
𝑢
∆−1 ∆+1
∆−1 ∆+1
∆−1 ∆+1
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 1: For any 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , 2 ∉
𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣) and ∆ + 1 ∉ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣).
• (pf):Suppose ∆ + 1 ∈ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣)
𝑣
∆+1 ∆+1
𝑢
∆−1 ∆+1
∆−1 ∆+1
∆−1 ∆+1
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 1: For any 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , 2 ∉
𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣) and ∆ + 1 ∉ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣).
• (pf):Suppose ∆ + 1 ∈ 𝐴𝑓 (𝑢) ∩ 𝐴𝑓 (𝑣)
𝑣
∆+1 ∆−1
𝑢
∆+1 ∆−1
∆+1 ∆−1
∆+1 ∆−1
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 2:
• If there exists a 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) ,
such that 𝐴𝑓 𝑢 = 𝐴𝑓 𝑣 = 3 , then there exists a
𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 ′ of 𝐻(3) , such that 2 ∈ 𝐴𝑓′ 𝑢
and 3 ∈ 𝐴𝑓′ 𝑣 .
Lemma 2:
𝑊: 𝑢 = 𝑢1 𝑢2 … 𝑢𝑙 𝑖𝑠 𝑎 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑛𝑔 −
𝑒 − 2, 𝑒; 𝑒 + 1, 𝑒 − 1 − 𝑡𝑟𝑎𝑖𝑙 for some 4 ≤ 𝑒 ≤ ∆
𝑢
𝑒−1
𝑒−2
𝑒+1
𝑒
𝑒
𝑒−2
𝑒+1
𝑒−1
𝑒+1
𝑒−2
𝑒
𝑒
𝑒−2
𝑒+1
𝑒−1
𝑒−2
𝑒+1
𝑒
𝑒
𝑒+1
𝑒−1
𝑢
𝑒−1
𝑒−1
𝑒−1
𝑒−2
𝑒+1
𝑒
𝑢𝑙
∵ 𝑒 + 1 ∈ 𝐴𝑓 (𝑢𝑙 )
𝑒−2
𝑢𝑙
∵ 𝑒 − 2 ∈ 𝐴𝑓 (𝑢𝑙 )
Lemma 2:
𝑊: 𝑢 = 𝑢1 𝑢2 … 𝑢𝑙 𝑖𝑠 𝑎 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑛𝑔 −
𝑒 − 2, 𝑒; 𝑒 + 1, 𝑒 − 1 − 𝑡𝑟𝑎𝑖𝑙 for some 4 ≤ 𝑒 ≤ ∆
𝑢
𝑒−1
𝑒−2
𝑒+1
𝑒
𝑒
𝑒−2
𝑒−1
𝑒+1
𝑒−1
𝑒−2
𝑎
𝑒−1
𝑒+1
𝑒−2
𝑒
𝑒
𝑒−2
𝑒+1
𝑒−1
𝑒−1
𝑒−2
𝑒+1
𝑒
𝑢𝑙
∵ (𝑒 − 1) − 𝑎 ≥ 2, 𝑎 ∉ 𝑒 − 2, 𝑒 − 1, 𝑒
𝑢
𝑒
𝑒+1
𝑏
𝑢𝑙
∵ 𝑒 − 𝑏 ≥ 2, 𝑏 ∉ 𝑒 − 1, 𝑒, 𝑒 + 1
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 2:
• If there exists a 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) ,
such that 𝐴𝑓 𝑢 = 𝐴𝑓 𝑣 = 3 , then there exists a
𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 ′ of 𝐻(3) , such that 2 ∈ 𝐴𝑓′ 𝑢
and 3 ∈ 𝐴𝑓′ 𝑣 .
𝑣
𝑢
3
3
2
4
5
3
2
4
2
3
5
4
2
3
5
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
• Case 3:
• If there exists a 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , such
that 2 ∈ 𝐴𝑓 (𝑢)(𝑟𝑒𝑠𝑝. Δ + 1 ∈ 𝐴𝑓 𝑢 ) and
3 ∈ 𝐴𝑓 (𝑣)(𝑟𝑒𝑠𝑝. Δ ∈ 𝐴𝑓 𝑣 ) ,then there exists a 𝜆 −
𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 ′ of 𝐻(3) , such that
4∈
𝐴𝑓′ 𝑢 (𝑟𝑒𝑠𝑝. Δ − 1 ∈ 𝐴𝑓′ 𝑢 ) and 3 ∈
𝑢
𝑣
𝐴𝑓′ 𝑣 (𝑟𝑒𝑠𝑝. Δ ∈ 𝐴𝑓′ 𝑣 ).
3
2
3
4
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺 3 = Δ + 1.
pf：
• Case 4:
• If there exists a 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 of 𝐻(3) , such that
𝐴𝑓 𝑢 = 𝐴𝑓 𝑣 = 𝑎 for some 𝑎, 4 ≤ 𝑎 ≤ ∆, then there
exists a 𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑓 ′ of 𝐻(3) , such that 𝐴𝑓 𝑢 ∪
𝐴𝑓 𝑣 = 𝑎, 𝑎 + 1 .
𝑢
𝑣
𝑎
𝑎
𝑎+1
𝑎
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
𝑣
𝑢
𝑎+1 𝑎
𝑎-1
𝑎-1 𝑎+1
𝑎
𝑎+2 𝑎
𝑎-1 𝑎+1
Theorem 4:
• If G is a graph with ∆(𝐺) ≥ 4, then 𝜆 𝐺
3
= Δ + 1.
pf：
𝑣
𝑢
𝑎+1 𝑎-1
𝑎
𝑎+2
𝑎+1 𝑎-1
𝑎 𝑎+2
Example:
• 𝜆 𝐺 3 = 4 for a graph G with ∆ 𝐺 = 3, but 𝐺(3) is not
𝜆 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡.
𝒖𝟏
0
4
2
1
4
3
𝒖𝟒 3
1 𝒖𝟑
1
3
0
0
2
4
𝒖𝟐
𝐺 = 𝐾4 − 𝑒 for some 𝑒 ∈ 𝐸(𝐾4 )
Theorem 5:
• If G is 3-regular, then 𝜆(𝐺(3) )=4 if and only if 𝑉(𝐺) can be
partitioned into two set 𝑆1 and 𝑆2 , so that 𝑆1 = 𝑆2 , and
𝑀 = 𝑢𝑣: 𝑢𝑣 ∈ 𝐸 𝐺 , 𝑎𝑛𝑑 𝑢 ∈ 𝑆1 , 𝑣 ∈ 𝑆2 is a perfect
matching in G.
• pf：
𝑺𝟏
𝑺𝟐
Theorem 5:
• If G is 3-regular, then 𝜆(𝐺(3) )=4 if and only if 𝑉(𝐺) can be
partitioned into two set 𝑆1 and 𝑆2 , so that 𝑆1 = 𝑆2 , and
𝑀 = 𝑢𝑣: 𝑢𝑣 ∈ 𝐸 𝐺 , 𝑎𝑛𝑑 𝑢 ∈ 𝑆1 , 𝑣 ∈ 𝑆2 is a perfect
matching in G.
• pf：
4
2
𝑺𝟏
0
2 4
0 2
4 0
0
4
2
0
4
2 4
0
2
2 0 4
2
0
4
𝑺𝟐
Theorem 5:
• If G is 3-regular, then 𝜆(𝐺(3) )=4 if and only if 𝑉(𝐺) can be
partitioned into two set 𝑆1 and 𝑆2 , so that 𝑆1 = 𝑆2 , and
𝑀 = 𝑢𝑣: 𝑢𝑣 ∈ 𝐸 𝐺 , 𝑎𝑛𝑑 𝑢 ∈ 𝑆1 , 𝑣 ∈ 𝑆2 is a perfect
matching in G.
• pf：
4
2
𝑺𝟏
0
0 2
2 4
4 0
0
2 4
3
3
3
3
1
1
1
1
4
2
4
0
0
2 0 4
2
2
0
4
𝑺𝟐
𝐿 2,1 − 𝑙𝑎𝑏𝑒𝑙𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟 of 𝐺(ℎ)
Definition:
• Given a graph G, a spanning subgraph 𝐹 of 𝐺 is a
factor of G.
• A set of factors 𝐹1 𝐹2 … 𝐹𝑘 of G is called a
𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 of G if 𝐸(𝐺) can be represented as an
edge-disjoint union of factors 𝐹1 𝐹2 … 𝐹𝑘 .
Lemma:
• Given a graph G with ∆ 𝐺 = ∆, there exist a
factorization 𝐹1 𝐹2 … 𝐹𝑘 of G, such that every vertex in
each 𝐹𝑖 has degree at most 2.
Theorem 6:
• If G is a graph with ∆(𝐺) ≥ 5, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 3 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
ℎ
= Δ + 1.
Theorem 7:
• If G is a graph with ∆(𝐺) ≥ 4, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 4 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
ℎ
= Δ + 1.
• Example:
∆= 4
Theorem 7:
• If G is a graph with ∆(𝐺) ≥ 4, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 4 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
ℎ
= Δ + 1.
• Example:
∆= 4
𝐹1
𝐹2
Theorem 7:
• If G is a graph with ∆(𝐺) ≥ 4, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 4 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
= Δ + 1.
ℎ
• Example:
0
∆= 4
2
3
5
1
3
4
2
0
2 5
3
0
2 5
𝐹1
3
0
2 5
3
0
Theorem 7:
• If G is a graph with ∆(𝐺) ≥ 4, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 4 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
ℎ
= Δ + 1.
• Example:
0
∆= 4
5
4
1
1
4
5
0
0
𝐹2
Theorem 7:
• If G is a graph with ∆(𝐺) ≥ 4, and ℎ is a function from
𝐸(𝐺) → ℕ so that ℎ(𝑒) ≥ 4 for all 𝑒 ∈ 𝐸(𝐺) ,then
𝜆 𝐺
= Δ + 1.
ℎ
• Example:
0
𝜆 𝐺ℎ =∆+1
∆= 4
2
4
5
3
5
1
1
3
1
5
0
2 5
3
0
4
2
4
2 5
3
0
2 5
3
0
Thanks for your listening!