L(p, q)- labelings of subdivision of graphs

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)-labelingof G： • A k-L(p,q)-labelingis 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 ： 4-L(2,1)-labeling 6-L(2,1)-labeling

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：

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 :

Lü • Lüstudied 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 .

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 0 a c d b

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

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

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

Lemma 1: a

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

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

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 .

for some Lemma 2:

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 .

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 .

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 . +1

Theorem 4: • If G is a graph with , then . pf： +1 -1 -1 +1 +2 +1 -1

Theorem 4: • If G is a graph with , then . pf： +1 -1 +2 +1 -1 +2

Example: • for a graph G with , but is not . 0 4 2 4 1 3 1 3 1 3 0 0 2 4 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：

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： 4 2 0 2 0 0 0 4 2 2 4 4 0 2 4 4 2 4 4 2 0 0 2 0

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： 4 2 0 2 0 0 0 4 2 2 4 4 3 3 3 3 1 1 1 1 4 4 2 4 4 2 0 0 2 0 2 0

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:

Theorem 7: • If G is a graph with , and is a function from so that for all ,then . • Example: 0 2 3 5 1 3 4 2 0 3 3 2 2 2 0 5 5 0 0 3 5

Theorem 7: • If G is a graph with , and is a function from so that for all ,then . • Example: 0 5 4 1 1 4 5 0 0

Theorem 7: • If G is a graph with , and is a function from so that for all ,then . • Example: 0 2 3 4 5 5 1 1 1 3 4 2 5 4 3 3 2 2 2 5 5 3 5 0 0 0 0

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L(p, q)- labelings of subdivision of graphs