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Optimal 2-pebbling number

Optimal 2-pebbling number. Hung-Hsing Chiang Department 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｝

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Optimal 2-pebbling number

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  1. Optimal 2-pebbling number Hung-Hsing Chiang Department of Mathematics Chung Yuan Christian University Advisor：Chin-Lin Shiue and Mu-Ming Wong

  2. Outline • Definetion • Preliminary • Main result

  3. 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 .

  4. 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.

  5. 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. 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)

  7. 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. G= Cn

  8. Preliminaries Definetion 2.1 Let x be a nonnegative integer. We define and for each i＞0

  9. Lemma 2.2 For each positive integer i Moreover if (x mod 2i＋y mod 2i ) ＜ 2i

  10. Example If i =1, x is anonnegative integer and y=2 then

  11. 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

  12. 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) .

  13. 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)

  14. 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)

  15. Lemma 2.3 Let δ be a distribution of Cn such thatδ(u) ≦2 for each u in Cnand let path P be a subgraph of Cn with an endvertex v .Then δP(v) ≦δ(v) +1

  16. Corollary 2.4 Let δ be a distribution of Cn such thatδ(u) ≦2 for each u in Cnand 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)

  17. The optimal 2-pebbling number of Cn • Proposition3.1

  18. Lemma3.2 There exists an optimal 2-pebbling δ﹡of Cn such thatδ﹡(v) ≦2 for each v ∊V(Cn).

  19. 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)

  20. 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…uksuch that δ(ui)=δ(uk)=2 and δ(u)=1 for each vertex u≠ui ,uj ,uk inP.

  21. The prove of Lemma3.3 1. There is a path P= uiui+1…ujuj+1…uksuch that δ(ui)=δ(uk)=2 and δ(u) ≦1 for each vertex u≠ui ,uj ,uk inP . 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

  22. 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.

  23. Thanks for your attention

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