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Optimal 2-Pebbling Numbers for Cycle Graphs: Analysis and Results

This study explores the concept of optimal 2-pebbling numbers in cycle graphs, focusing on their definitions, preliminary results, and main findings. We define the distribution of pebbles on vertices in a graph and introduce the concept of 2-pebbling moves. The primary aim is to determine the minimum number of pebbles required for an α-pebbling of cycle graphs. Through various examples and propositions, including bounds on pebbling numbers for specific distributions, we unveil significant properties and conclusions that enhance our understanding of pebbling in graph theory.

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Optimal 2-Pebbling Numbers for Cycle Graphs: Analysis and Results

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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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