Optimal 2-pebbling number

Hung-Hsing ChiangDepartment of Mathematics

Chung Yuan Christian University Advisor ： Chin-Lin Shiue and Mu-Ming

Wong

Outline

Definetion Preliminary Main result

Definitions

A distribution δ of pebble of G is a function ： V(G) →N∪ ｛ 0 ｝ δ （ v ） be the number of pebbles

distribution to v.

δ （ G ） be the number of pebbles that be distributed vertices of G .

GpvGVv

1.Pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex.

2.If a distribution δ of pebbles lets us move at least one pebbles to each vertex v by applying pebbling moves repeatly(if necessary), then δ is called a pebbling of G.

3. δG(v) denote the maximum number of pebbles which can be moved to v by applying pebbling moves on G.

4.A pebbling type αof G is a mapping from

V(G) onto N∪ ｛ 0 ｝

5.A distribution δ is called an α-pebbling if whenever we choose a target vertex v, we can move α(v) pebbles to v by applying pebbling moves.

6.An ι-pebbling of G is an α-pebbling of G for which α(v) = ιfor each v in G, where ιis a positive integer.

7.The optimal α pebbling number of G f’α (G) is the minimum number of pebbles used in an α-pebbling of G.

Note ： optimal 1 pebbling number of G is optimal pebbling number of G ,f’ (G)

Example

δ(u1)=1 δ(u2)=0 δ(u3)=3

δ(u4)=2 δ(u5)=2 δ(G)=8

δG(u1)=3 δG(u2)=3 δG(u3)=4

δG(u4)=4 δG(u5)=3

δ is a 3-pebbling of G.

u 5

1

0

32

2

u 4 u 3

u 2

u 1

G= Cn

Preliminaries

Definetion 2.1 Let x be a nonnegative integer. We define

and for each i ＞ 0

21

xx i

i

xx 0

Lemma 2.2

For each positive integer i Moreover

if (x mod 2i ＋ y mod 2i ) ＜ 2i

1 yxyx iii

yxyx iii

Example

If i =1, x is a nonnegative integer

and y=2 then

1 1 12 2x x

Cycle graph Cn

Cn =u0u1…un-1u0

where subscript is modulo n Two path P+(ui , uj)=uiui+1…uj and

P-(ui , uj)=uiui-1…u1unun-1…uj

Let δ be a distribution of Cn.. If we let δ+ be a distribution of P+(ui , uj) such that δ+(ui )=x ≦δ(ui ) and δ+(v)= δ(v) for each v≠ui in P+(ui , uj) then

m+(δ,x, ui , uj) denotes

where P+= P+(ui , uj) .

The similarly m-(δ,y, ui , uj) = where P-= P-(ui , uj) .

j jPu u

j jPu u

Fact 1

(i)We can choose some vertives w of Cn and two integers x≧2 and y ≧ 2, x+y=δ(w), such that

δCn(u) = m+(δ,x,w,u)+ m-(δ,y,w,u)+ δ(u) (ii)We can choose some vertives w of Cn with δ(w) ≧2 such that

δCn(u) = m+(δ, δ(w) ,w,u)+ δ(u) or

δCn(u) = m-(δ, δ(w) ,w,u)+ δ(u)

Fact 1

(iii)We can choose two vertives ui

and uj in V(Cn) where δ(ui)≧2 ,δ(uj) ≧ 2 and P+(ui , u) ∩ P-(uj, u)= ｛ u ｝ , such that δCn(u) = m+(δ, δ(ui), ui,u)+ m-(δ, δ(uj) ,uj,u)+ δ(u)

(iv)δCn(u) = δ(u)

Lemma 2.3

Let δ be a distribution of Cn such that δ(u) ≦2 for each u in Cn and let path P be a subgraph of Cn with an endvertex v .Then δP(v) ≦δ(v) +1

Corollary 2.4

Let δ be a distribution of Cn such that δ(u) ≦2 for each u in Cn and let v and w be two distinct vertices of Cn.. Then m+(δ,x,w,v)≦1 and m-(δ,y,w,v)≦1 for any two nonnegative integers x and y with x+y=δ(w)

The optimal 2-pebbling number of Cn

Proposition3.1 nCf n '

2

u20

2

0

11

1

1

1

1

11

2

u4

u3

u1

otherwise 1

or if 2

or if 0

42

31

iii

iii

ui

Lemma3.2

There exists an optimal 2-pebbling δ﹡

of Cn such that δ﹡(v) ≦2 for each v ∊V(Cn).

The prove of Lemma3.2

1.Let δ be an optimal 2-pebbling of Cn and δ(uk)=max｛ δ(v) |v ∊ V(Cn) ｝ .

2.Pk=uk-muk-m+1…uk…uk+l-1uk+l where δ(uk-m)=

δ(uk+l)=0 and δ(v) ＞ 0 for each internal vertex of Pk.

3.Let δ’ be a distribution of Cn defined by δ’(uk)= δ(uk)-2, δ’(uk-m)= δ(uk-m)+1, δ’(uk+l)= δ(uk+l)+1 and δ’(v)= δ(v) for v∊V(Cn)∖ ｛ uk-m , uk , uk+l｝ .

4.Claim 1. δ’Cn(u) ≧2 for each vertex u∊V(Pk)

Claim 2. δ’Cn(u) ≧2 for each vertex u∊ V(Cn) ∖ (Pk)

Lemma3.3

Let δ is an optimal 2-pebbling of Cn

with δ(u) ≦2 for each u in Cn . If δ(uj)=0 then exists a path P= uiui+1…ujuj+1…uk such that δ(ui)=δ(uk)=2 and δ(u)=1 for each vertex u≠ui ,uj ,uk in P.

The prove of Lemma3.3

1. There is a path P= uiui+1…ujuj+1…uk such that δ(ui)=δ(uk)=2 and δ(u) ≦1 for each vertex u≠ui ,uj ,uk in P .

2. Suppose there exists a vertex ul for i ＜ l ＜ j such that δ(ul) =0

3. By Fact1 and Corollary 2.4 δCn(uj) ≦1

Proposition3.4

Proof: Let δ is an optimal 2-pebbling of Cn with

δ(u) ≦2 for each u in Cn . Assume δ(Cn) =k ≦n-1. Let A and B are two subsets of V(Cn) where

A= ｛ u ∊ V(Cn) |δ(u)=0 ｝ and B= ｛ u ∊ V(Cn) |δ(u)=2 ｝ . Since δ(Cn)≦n-1 and δ(u) ≦2 for each u ∊ V(Cn), |A|≧|B|+1. But |A|≦|B| by Lemma3.3, it is a contradiction. Hence δ(Cn)≧n.

2

'nf C n

Thanks for your attention