A pebbling move on a graph removes two pebbles at a vertex and adds

  one pebble at an adjacent vertex. Rubbling is a version of pebbling

  where an additional move is allowed. In this new move, one pebble

  each is removed at vertices $v$ and $w$ adjacent to a vertex $u$,

  and an extra pebble is added at vertex $u$. A vertex is reachable

  from a pebble distribution if it is possible to move a pebble to

  that vertex using rubbling moves.  The optimal pebbling (rubbling) number is

  the smallest number $m$ needed to guarantee a pebble distribution of

  $m$ pebbles from which any vertex is reachable using pebbling (rubbling) moves. 

	We determine the optimal rubbling number of ladders ($P_n\square P_2$), prisms

  ($C_n\square P_2$) and M\"oblus-ladders. We also give upper and lower 

	bounds for the optimal pebbling and rubbling numbers of large grids ($P_n\square P_n$)

Győri, Ervin

Katona, Gyula Y.

Papp, László F.

Repository of the Academy's Library

Optimal pebbling of gridsErvin Gyo˝ri1, Gyula Y. Katona2,3, and La´szlo´ Papp21Alfre´d Re´nyi Institute of Mathematics, Hungarian Academy of Sciences2Department of Computer Science and Information Theory, Budapest University of Technology andEconomics3MTA-ELTE Numerical Analysis and Large Networks Research Group1 IntroductionGraph pebbling has its origin in number theory. It is a model for the transportation of re-sources. Starting with a pebble distribution on the vertices of a simple connected graph, apebbling move removes two pebbles from a vertex and adds one pebble at an adjacent vertex.We can think of the pebbles as fuel containers. Then the loss of the pebble during a move isthe cost of transportation. A vertex is called reachable if a pebble can be moved to that vertexusing pebbling moves. There are several questions we can ask about pebbling. One of them is:How can we place the smallest number of pebbles such that every vertex is reachable (optimalpebbling number)? For a comprehensive list of references for the extensive literature see thesurvey papers [4, 5, 6]. Results on special grids can be found in [2] where the authors showthat piopt(PnP2) = piopt(CnP2) = n apart from a few smaller case, and in [11] the authorgave upper bounds for the optimal pebbling number of various grids.In the present paper we give better upper and lower bounds for the optimal pebbling numbersof large grids (PnPn).Graph rubbling is an extension of graph pebbling. In this version, we also allow a move thatremoves a pebble each from the vertices v and w that are adjacent to a vertex u, and addsa pebble at vertex u. The basic theory of rubbling and optimal rubbling is developed in [1].The rubbling number of complete m-ary trees are studied in [3], while the rubbling numberof caterpillars are determined in [10]. In [7] the authors gives upper and lower bounds for therubbling number of diameter 2 graphs.In the present paper we determine the optimal rubbling number of ladders (PnP2), prisms(CnP2) and Mo¨blus-ladders. We also give upper and lower bounds for the optimal rubblingnumbers of large grids (PnPn).2 DefinitionsThroughout the paper, let G be a simple connected graph. We use the notation V (G) for thevertex set and E(G) for the edge set. A pebble function on a graph G is a function p : V (G)→ Zwhere p(v) is the number of pebbles placed at v. A pebble distribution is a nonnegative pebblefunction. The size of a pebble distribution p is the total number of pebbles∑v∈V (G) p(v). Wesay that a vertex v is occupied if p(v) > 1, else it is unoccupied.Consider a pebble function p on the graph G. If {v, u} ∈ E(G) then the pebbling move(v, v→u) removes two pebbles at vertex v, and adds one pebble at vertex u to create a newpebble function p′, so p′(v) = p(v)− 2 and p′(u) = p(u) + 1. If {w, u} ∈ E(G) and v 6= w, thenthe strict rubbling move (v, w→u) removes one pebble each at vertices v and w, and adds onepebble at vertex u to create a new pebble function p′, so p′(v) = p(v) − 1, p′(w) = p(w) − 1and p′(u) = p(u) + 1.A rubbling move is either a pebbling move or a strict rubbling move. A rubbling sequenceis a finite sequence T = (t1, . . . , tk) of rubbling moves. The pebble function obtained from thepebble function p after applying the moves in T is denoted by pT . The concatenation of the rub-bling sequences R = (r1, . . . , rk) and S = (s1, . . . , sl) is denoted by RS = (r1, . . . , rk, s1, . . . , sl).A rubbling sequence T is executable from the pebble distribution p if p(t1,...,ti) is nonnegativefor all i. A vertex v of G is reachable from the pebble distribution p if there is an executablerubbling sequence T such that pT (v) ≥ 1. p is a solvable distribution when each vertex isreachable. All the above notions are defined for pebbling as well, just we restrict ourselves topebbling moves.The optimal pebbling piopt(G) and rubbling number %opt(G) of a graph G is the size ofa distribution with the least number of pebbles from which every vertex is reachable usingpebbling/rubbling moves. For large graphs it is better to consider the ratio of the optimalpebbling or rubbling number and the number of the vertices of the graph. So the Opti-mal Pebbling Density is OPD(G) = piopt(G)/|V (G)| and the Optimal Rubbling Density isORD(G) = %opt(G)/|V (G)|.Let G and H be simple graphs. Then the Cartesian product of graphs G and H is the graphwhose vertex set is V (G) × V (H) and (g, h) is adjacent to (g′, h′) if and only if g = g′ and(h, h′) ∈ E(H) or if h = h′ and (g, g′) ∈ E(G). This graph is denoted by GH.Pn and Cn denotes the path and the cycle containing n distinct vertices, respectively. Wecall PnP2 a ladder and CnP2 a prism, and Pn1Pn2 in general a grid. It is clear that theprism can be obtained from the ladder by joining the 4 endvertices by two edges to form twovertex disjoint Cn subgraphs. If the four endvertices are joined by two new edges in a switchedway to get a C2n subgraph, then a Mo¨bius-ladder is obtained.3 Optimal rubbling number of the ladder, the n-prism andMo¨bius-ladderOur main result is the following formula for the optimal rubbling number of ladders:Theorem 1. Let n = 3k + r such that 0 ≤ r < 3 and n, r ∈ N, so k = ⌊n3 ⌋.%opt(PnP2) =1 + 2k if r = 0,2 + 2k if r = 1,2 + 2k if r = 2.So ORD(PnP2) ≈ 13 .To show that the above values are upper bounds for %opt(PnP2) it is enough to give a solv-able distribution. It is not too hard to show that these are really solvable. Such distributionsare shown on Figure 1.We also prove that this is best posibble.0 0 20 0002 00 200002010 11 0000 11 0100001010 11 0000 11 010003k+13k+23kFigure 1: Optimal distributions.4 Optimal pebbling and rubbling numbers of large gridsWe turn our attention to larger grids now, in the following we assume that n is large enough(say ≥ 100). Shiue [11] proved that the analogue of Graham’s conjecture for optimal pebblingis true: piopt(G1G2) ≤ piopt(G1)piopt(G2). Since in [9] it was proved that piopt(Pn) = d2n/3e,this implies that OPD(PnPn) ≤ 49 + o(1). In [12] the authors gave a construction showingthat OPD(PnPn) ≤ 413 + o(1). Our first result is better construction.Theorem 2.piopt(PnPn) ≤ 27n2 +O(n),so OPD(PnPn) ≤ 27 + o(1).We conjecture that this is a sharp bound. Applying the well known weight argument, it isfairly easy to obtain that OPD(PnPn) ≥ 19 . The authors in [12] claim OPD(PnPn) ≥ 16 .Unfortunately, we belive that their proof contains an error, may be it can be corrected easily,but we do not see how. However, they introduced an interesting notion: excess weight. Usingthis notion, but following a different approach we proved the following lower bound.Theorem 3. OPD(PnPn) ≥ 425 .For the optimal rubbling number of large grids we do not know any previous results. Wegive a construction to prove:Theorem 4.%opt(PnPn) ≤ 15n2 +O(n),so ORD(PnPn) ≤ 15 + o(1).We conjecture that this is a sharp bound. A simlilar argument to the one we used fo pebblingalso gives a nontrivial lower bound for the optimal rubbling number.Theorem 5. ORD(PnPn) ≥ 537 .5 AcknowledgmentThe author acknowledges support from OTKA (grant no.108947)References[1] Ch. Belford and N. Sieben, Rubbling and optimal rubbling of graphs, Discrete Math. 309no. 10,(2009) pp. 3436–3446.[2] D.P. Bunde, E. W. Chambers, D. Cranston, K. Milans, D. B. West, Pebbling and optimalpebbling in graphs J. Graph Theory 57 no. 3. (2008) pp. 215–238.[3] L. Danz, Optimal t-rubbling of complete m-ary trees, REU project report, University ofMinnesota Duluth, Department of Mathematics and Statistics, (2010).[4] G. Hurlbert, A survey of graph pebbling, Proceedings of the Thirtieth Southeastern Inter-national Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL,1999), vol. 139, (1999), pp. 41–64.[5] G. Hurlbert, Recent progress in graph pebbling, Graph Theory Notes N. Y. 49 (2005),pp. 25–37.[6] G. Hurlbert, Graph pebbling, In: Handbook of Graph Theory, Ed: J. L. Gross, J. Yellen,P. Zhang, Chapman and Hall/CRC, Kalamazoo (2013), pp. 1428– 1449.[7] G. Y. Katona, N. Sieben, Bounds on the Rubbling and Optimal Rubbling Numbers of GraphsElectronic Notes in Discrete Mathematics, Volume 38, (2011), pp. 487-492.[8] D. Moews, Pebbling graphs, J. Combin. Theory (B), 55 (1992) pp. 244–252.[9] L. Pachter, S.S. Hunter, B. Voxman, On pebbling graphs, Proceedings of the Twenty-sixthSoutheastern International Conference on Combinatorics, Graph Theory and Computing(Boca Raton, FL, 1995), Congr. Numer.,107, (1995)[10] L. Papp, Optimal rubbling numbers of graphs (in Hungarian), thesis, Budapest Universityof Technology and Economics, Department of Computer Science and Information Theory,(2010).[11] C. L. Shiue, Optimally pebbling graphs, Ph. D. Dissertation, Department of Applied Math-ematics, National Chiao Tung University, (1999) Hsin chu, Taiwan.[12] C. Xue, C. Yerger, Optimal Pebbling on Grids, manuscript (2014)

Optimal pebbling of grids

Optimal pebbling of gridsErvin Gyo˝ri∗Alfre´d Re´nyi Institute of MathematicsHungarian Academy of Sciences1053 Rea´ltanoda u. 13-15, Budapest, Hungarygyori.ervin@renyi.mta.huGyula Y. Katona†Department of Computer Science andInformation TheoryBudapest University of Technology andEconomics1117 Budapest, Mu˝egyetem rkpt. 2, HungaryandMTA-ELTE Numerical Analysis and LargeNetworks Research Groupkiskat@cs.bme.huLa´szlo´ Papp†Department of Computer Science andInformation TheoryBudapest University of Technology andEconomics1117 Budapest, Mu˝egyetem rkpt. 2, Hungarylazsa@cs.bme.huAbstract: A pebbling move on a graph removes two pebbles at a vertex and adds one pebbleat an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed.In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u,and an extra pebble is added at vertex u. A vertex is reachable from a pebble distributionif it is possible to move a pebble to that vertex using rubbling moves. The optimal pebbling(rubbling) number is the smallest number m needed to guarantee a pebble distribution of mpebbles from which any vertex is reachable using pebbling (rubbling) moves. We determinethe optimal rubbling number of ladders (PnP2), prisms (CnP2) and Mo¨blus-ladders. Wealso give upper and lower bounds for the optimal pebbling and rubbling numbers of largegrids (PnPn).Keywords: pebbling, rubbling, optimal pebbling1 IntroductionGraph pebbling has its origin in number theory. It is a model for the transportation of resources.Starting with a pebble distribution on the vertices of a simple connected graph, a pebbling move removestwo pebbles from a vertex and adds one pebble at an adjacent vertex. We can think of the pebbles as fuelcontainers. Then the loss of the pebble during a move is the cost of transportation. A vertex is calledreachable if a pebble can be moved to that vertex using pebbling moves. There are several questionswe can ask about pebbling. One of them is: How can we place the smallest number of pebbles suchthat every vertex is reachable (optimal pebbling number)? For a comprehensive list of references for theextensive literature see the survey papers [4, 5, 6]. Results on special grids can be found in [2] where∗Research is supported by the Hungarian National Research Fund (grant number 101536).†Research is supported by the Hungarian National Research Fund (grant number 108947).the authors show that piopt(PnP2) = piopt(CnP2) = n apart from a few smaller case, and in [11] theauthor gave upper bounds for the optimal pebbling number of various grids.In the present paper we give better upper and lower bounds for the optimal pebbling numbers of largegrids (PnPn).Graph rubbling is an extension of graph pebbling. In this version, we also allow a move that removesa pebble each from the vertices v and w that are adjacent to a vertex u, and adds a pebble at vertex u.The basic theory of rubbling and optimal rubbling is developed in [1]. The rubbling number of completem-ary trees are studied in [3], while the rubbling number of caterpillars are determined in [10]. In [7] theauthors gives upper and lower bounds for the rubbling number of diameter 2 graphs.In the present paper we determine the optimal rubbling number of ladders (PnP2), prisms (CnP2)and Mo¨blus-ladders. We also give upper and lower bounds for the optimal rubbling numbers of largegrids (PnPn).2 DefinitionsThroughout the paper, let G be a simple connected graph. We use the notation V (G) for the vertex setand E(G) for the edge set. A pebble function on a graph G is a function p : V (G) → Z where p(v) isthe number of pebbles placed at v. A pebble distribution is a nonnegative pebble function. The size of apebble distribution p is the total number of pebbles∑v∈V (G) p(v). We say that a vertex v is occupied ifp(v) > 1, else it is unoccupied.Consider a pebble function p on the graph G. If {v, u} ∈ E(G) then the pebbling move (v, v→u)removes two pebbles at vertex v, and adds one pebble at vertex u to create a new pebble function p′,so p′(v) = p(v) − 2 and p′(u) = p(u) + 1. If {w, u} ∈ E(G) and v 6= w, then the strict rubbling move(v, w→u) removes one pebble each at vertices v and w, and adds one pebble at vertex u to create a newpebble function p′, so p′(v) = p(v)− 1, p′(w) = p(w)− 1 and p′(u) = p(u) + 1.A rubbling move is either a pebbling move or a strict rubbling move. A rubbling sequence is a finitesequence T = (t1, . . . , tk) of rubbling moves. The pebble function obtained from the pebble functionp after applying the moves in T is denoted by pT . The concatenation of the rubbling sequences R =(r1, . . . , rk) and S = (s1, . . . , sl) is denoted by RS = (r1, . . . , rk, s1, . . . , sl).A rubbling sequence T is executable from the pebble distribution p if p(t1,...,ti) is nonnegative for alli. A vertex v of G is reachable from the pebble distribution p if there is an executable rubbling sequenceT such that pT (v) ≥ 1. p is a solvable distribution when each vertex is reachable. All the above notionsare defined for pebbling as well, just we restrict ourselves to pebbling moves.The optimal pebbling piopt(G) and rubbling number %opt(G) of a graph G is the size of a distributionwith the least number of pebbles from which every vertex is reachable using pebbling/rubbling moves.For large graphs it is better to consider the ratio of the optimal pebbling or rubbling number and thenumber of the vertices of the graph. So the Optimal Pebbling Density is OPD(G) = piopt(G)/|V (G)| andthe Optimal Rubbling Density is ORD(G) = %opt(G)/|V (G)|.Let G and H be simple graphs. Then the Cartesian product of graphs G and H is the graph whosevertex set is V (G)× V (H) and (g, h) is adjacent to (g′, h′) if and only if g = g′ and (h, h′) ∈ E(H) or ifh = h′ and (g, g′) ∈ E(G). This graph is denoted by GH.Pn and Cn denotes the path and the cycle containing n distinct vertices, respectively. We call PnP2a ladder and CnP2 a prism, and Pn1Pn2 in general a grid. It is clear that the prism can be obtainedfrom the ladder by joining the 4 endvertices by two edges to form two vertex disjoint Cn subgraphs.If the four endvertices are joined by two new edges in a switched way to get a C2n subgraph, then aMo¨bius-ladder is obtained.3 Optimal rubbling number of the ladder, the n-prism and Mo¨bius-ladderOur main result is the following formula for the optimal rubbling number of ladders:Theorem 1 Let n = 3k + r such that 0 ≤ r < 3 and n, r ∈ N, so k = ⌊n3 ⌋.%opt(PnP2) = 1 + 2k if r = 0,2 + 2k if r = 1,2 + 2k if r = 2.So ORD(PnP2) ≈ 13 .To show that the above values are upper bounds for %opt(PnP2) it is enough to give a solvabledistribution. It is not too hard to show that these are really solvable. Such distributions are shown onFigure 1.0 0 20 0002 00 200002010 11 0000 11 0100001010 11 0000 11 010003k+13k+23kFigure 1: Optimal distributions.We also need to prove that it is a lower bound. This is done by induction on n, we give a summeryof the proof.Consider a p optimal distribution on PnP2. Choose an appropriate R = P3P2 subgraph, deletethe vertices of R and reconnect the remaining two parts to obtain GR = Pn−3P2, called the reducedgraph, see Fig 2.Now construct a solvable p′ distribution for the new Pn−3P2 graph in the following way: p inducesa distribution on the vertices which we have not deleted. Place p(v) pebbles to all v ∈ V (G)\V (R). (InBBAABBllxAxArrFigure 2: Deleting a P3P2 subgraph.some cases we need to swap the pebbles on the rungs.) Finally, distribute and place Rp − 2 pebbles atvertices A, A, B and B in an appropriate way so that the new distribution on Pn−3P2 is solvable. Ouraim is to show that it is always possible to find such a new distribution. This is proved in several lemmasand case analysis. These will imply%opt(PnP2) ≥ %opt(Pn−3P2) + 2.It is easy to see that this implies the theorem if we show that the theorem holds for n = 1, 2, 3.Lemma 2%opt(P2) = 2, %opt(P2P2) = 2, %opt(P3P2) = 3.Proof: The optimal distributions are shown in Fig. 3. It is an easy exercise to check that thesedistributions are optimal.11101 00 11001Figure 3: Optimal distributions of P2, P2P2 and P3P2.In the rest of the paper we also determine the optimal pebbling number for the n-prism (and theMo¨bius-ladder).Theorem 3 %opt(C3k−1P2) = %opt(P3k−2P2) = 2k, %opt(C3kP2) = %opt(P3k−1P2) = 2k,%opt(C3k+1P2) = %opt(P3kP2) = 2k + 1. Except: %opt(C3P2) = 3, %opt(C4P2) = 4.4 Optimal pebbling and rubbling numbers of large gridsWe turn our attention to larger grids now, in the following we assume that n is large enough (say ≥ 100).Shiue [11] proved that the analogue of Graham’s conjecture for optimal pebbling is true: piopt(G1G2) ≤piopt(G1)piopt(G2). Since in [9] it was proved that piopt(Pn) = d2n/3e, this implies that OPD(PnPn) ≤49 + o(1). In [12] the authors gave a construction showing that OPD(PnPn) ≤ 413 + o(1). Our firstresult is better construction.Theorem 4piopt(PnPn) ≤ 27n2 +O(n),so OPD(PnPn) ≤ 27 + o(1).We conjecture that this is a sharp bound. Applying the well known weight argument, it is fairlyeasy to obtain that OPD(PnPn) ≥ 19 . The authors in [12] claim OPD(PnPn) ≥ 16 . Unfortunately,we believe that their proof contains an error, may be it can be corrected easily, but we do not see how.However, they introduced an interesting notion: excess weight. Using this notion, but following a differentapproach we proved the following lower bound.Theorem 5 OPD(PnPn) ≥ 213 .For the optimal rubbling number of large grids we do not know any previous results. We give aconstruction to prove:Theorem 6%opt(PnPn) ≤ 15n2 +O(n),so ORD(PnPn) ≤ 15 + o(1).We conjecture that this is a sharp bound. A similar argument to the one we used for pebbling alsogives a nontrivial lower bound for the optimal rubbling number.Theorem 7 ORD(PnPn) ≥ 537 .References[1] Ch. Belford and N. Sieben, Rubbling and optimal rubbling of graphs, Discrete Math. 309 no. 10,(2009)pp. 3436–3446.[2] D.P. Bunde, E. W. Chambers, D. Cranston, K. Milans, D. B. West, Pebbling and optimal pebbling ingraphs J. Graph Theory 57 no. 3. (2008) pp. 215–238.[3] L. Danz, Optimal t-rubbling of complete m-ary trees, REU project report, University of MinnesotaDuluth, Department of Mathematics and Statistics, (2010).[4] G. Hurlbert, A survey of graph pebbling, Proceedings of the Thirtieth Southeastern InternationalConference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1999), vol. 139,(1999), pp. 41–64.[5] G. Hurlbert, Recent progress in graph pebbling, Graph Theory Notes N. Y. 49 (2005), pp. 25–37.[6] G. Hurlbert, Graph pebbling, In: Handbook of Graph Theory, Ed: J. L. Gross, J. Yellen, P. Zhang,Chapman and Hall/CRC, Kalamazoo (2013), pp. 1428– 1449.[7] G. Y. Katona, N. Sieben, Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs ElectronicNotes in Discrete Mathematics, Volume 38, (2011), pp. 487-492.[8] D. Moews, Pebbling graphs, J. Combin. Theory (B), 55 (1992) pp. 244–252.[9] L. Pachter, S.S. Hunter, B. Voxman, On pebbling graphs, Proceedings of the Twenty-sixth South-eastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL,1995), Congr. Numer.,107, (1995)[10] L. Papp, Optimal rubbling numbers of graphs (in Hungarian), thesis, Budapest University of Tech-nology and Economics, Department of Computer Science and Information Theory, (2010).[11] C. L. Shiue, Optimally pebbling graphs, Ph. D. Dissertation, Department of Applied Mathematics,National Chiao Tung University, (1999) Hsin chu, Taiwan.[12] C. Xue, C. Yerger, Optimal Pebbling on Grids, manuscript (2014)

Optimal pebbling of grids

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