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

• with

• Reference

Introduction

Definition:• L(p,q)-labeling of G： • 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 , is the smallest number k such that G has a k-L(p,q)- labeling.• .

Example:• L(2,1)-labeling of ： 0 2

6 4

6-L(2,1)-labeling

0

4 1

3

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 .

• They conjectured that for any graph G with .

Gonçalves (2005)• with .

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：

3C

Definition:• Given a graph G and a function from to , the of G,

denoted by , is the graph obtained from G by replacing each edge in G with a path , where .

• If for all , we use to replace .• Example :

𝐺(3) 𝐺(5)

𝐺=𝐶 4

Lü• Lü studied the of and conjectured that for and

graph with maximum degree .

Karst • Karst et al. studied the of and gave upper bound

for (where .• Karst et al. showed that for any graph G with . From this, we have for all graph G with and

with is even and .

Extend of subdivisions of graphs.• for any graph G with .•

if and is a function from to so that for all , or if and is a function from to so that for all .

of for

Georges and Mauro • They studied the L(p,q)-labeling numbers of

paths and cycles and gave the following results.

Theorem 1:• For all ,

Theorem 2:• For all then • If , then

Theorem 3:• If , then

where

.

of

Definition:• An of is said to be if for all .

• The of in G, denoted , is defined by .

• If is a of and , then we use to denoted the set .

• An of is said to be a of if is and .

Definition:• Let be an of G. For , a trail in is called in

corresponding to if

0 0 0a c b d

Theorem 4:• If G is a graph with , then . pf： 𝑢 𝑣

𝐺∗ is minimum

Theorem 4:• If G is a graph with , then . pf： 𝑢 𝑣

𝐻

Theorem 4:• If G is a graph with , then . pf：• Case 1: For any of , and .

Lemma 1:

𝑢 𝑢Δ −1

𝑢𝑙

𝑢𝑙

∵ Δ+1∈𝐴 𝑓 (𝑢𝑙) ∵𝑎必定≤ ∆ − 3a

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

Δ −1

Δ −1

Theorem 4:• If G is a graph with , then . pf：• Case 1: For any of , and .• Suppose )𝑣 𝑢∆+1∆+1

∆+1 ∆+1 ∆+1∆ − 1 ∆ − 1 ∆ − 1

Theorem 4:• If G is a graph with , then . pf：• Case 1: For any of , and .• (pf):Suppose )𝑣 𝑢∆+1∆+1

∆+1 ∆+1 ∆+1∆ − 1 ∆ − 1 ∆ − 1

Theorem 4:• If G is a graph with , then . pf：• Case 1: For any of , and .• (pf):Suppose )𝑣 𝑢∆+1

∆+1 ∆+1 ∆+1∆ − 1 ∆ − 1 ∆ − 1∆ − 1

Theorem 4:• If G is a graph with , then . pf：• Case 2:• If there exists a of , such that , then there exists a of , such that and .

Lemma 2:

𝑢 𝑢

𝑒+1

𝑢𝑙

𝑒− 2

𝑒− 1

𝑒

𝑒− 2

𝑒

𝑒+1𝑒− 1

𝑒− 2

𝑒+1𝑒− 1

𝑒

∵𝑒+1∈𝐴 𝑓 (𝑢𝑙)

𝑒+1𝑒− 1

𝑒− 1𝑒+1

𝑒− 2

𝑒− 2

𝑒

𝑒

𝑢𝑙∵𝑒−2∈𝐴 𝑓 (𝑢𝑙)

𝑒

𝑒+1𝑒− 1

𝑒− 2

𝑒− 2𝑒

𝑒− 1𝑒+1

for some

Lemma 2:

𝑢 𝑢𝑒+1

𝑢𝑙

𝑒− 2

𝑒− 1

𝑒− 2

𝑒

𝑒+1𝑒− 1

𝑒− 2

𝑒− 1

𝑒

𝑎

∵|(𝑒−1)−𝑎|≥ 2 ,𝑎∉ {𝑒− 2 ,𝑒−1 ,𝑒 }

𝑒+1𝑒− 1

𝑒− 1𝑒+1

𝑒+1

𝑒− 2

𝑏

𝑒

𝑢𝑙

𝑒

𝑒+1𝑒− 1

𝑒− 2

𝑒− 2𝑒

𝑒

∵|𝑒−𝑏|≥ 2 ,𝑏∉ {𝑒−1 ,𝑒 ,𝑒+1 }

for some

Theorem 4:• If G is a graph with , then . pf：• Case 2:• If there exists a of , such that , then there exists a of , such that and .𝑣 𝑢

33 423542

2 3 3 55 4 2

Theorem 4:• If G is a graph with , then . pf：• Case 3: • If there exists a of , such that and ,then there exists a of , such that and .

𝑣 𝑢3 2

3 4

Theorem 4:• If G is a graph with , then . pf：• Case 4: • If there exists a of , such that for some , , then there exists a of , such that .

𝑣 𝑢𝑎 𝑎

+1 𝑎

Theorem 4:• If G is a graph with , then . pf：

𝑎+1 -1 +1 +2 𝑎 -1 +1

𝑣 𝑢

𝑎-1

Theorem 4:• If G is a graph with , then . pf：

𝑎+1 -1 +1+2 𝑎-1

𝑣 𝑢+2

Example:• for a graph G with , but is not . 𝒖𝟏

𝒖𝟒𝒖𝟑

𝒖𝟐

1

4

1

00

0

4

43

3

3

1

2

2

for some

Theorem 5:• If G is 3-regular, then =4 if and only if can be partitioned

into two set and , so that , and is a perfect matching in G.• pf：

𝑺𝟏

𝑺𝟐

Theorem 5:• If G is 3-regular, then =4 if and only if can be partitioned

into two set and , so that , and is a perfect matching in G.• pf：

𝑺𝟏

𝑺𝟐

4

4

0

2

4 4

444

400

0

2

222

222

0

0 00

Theorem 5:• If G is 3-regular, then =4 if and only if can be partitioned

into two set and , so that , and is a perfect matching in G.• pf：

𝑺𝟏

4

04 4 400

0

2

222

𝑺𝟐4 4

44222 0 0020

3 333

1 1 1 1

of

Definition:• Given a graph G, a spanning subgraph of is a factor of

G.• A set of factors of G is called a of G if can be

represented as an edge-disjoint union of factors .

Lemma:• Given a graph G with , there exist a factorization of G,

such that every vertex in each has degree at most 2.

Theorem 6:• If G is a graph with , and is a function from so that for all ,then .

Theorem 7:• If G is a graph with , and is a function from so that for all ,then .• Example:

∆=4

Theorem 7:• If G is a graph with , and is a function from so that for all ,then .• Example:

∆=4

𝐹 1 𝐹 2

Theorem 7:• If G is a graph with , and is a function from so that for all ,then .• Example:

∆=40

0 0 00 2 22

2

2

3

3

33

3

5 55

5 1

4

𝐹 1

Theorem 7:• If G is a graph with , and is a function from so that for all ,then .• Example:

∆=40

0 0

4

4

1 1

5

5

𝐹 2

Theorem 7:• If G is a graph with , and is a function from so that for all ,then .• Example:

∆=40

0 0 00

4

4

1 1

5

52

2

3

3

5 1

4

2 22 333 5 55

𝜆 (𝐺( h ))=∆+1

Thanks for your listening!