Abstract
AbstractGiven a distribution of pebbles on the vertices of a connected graphG, apebbling moveonGconsists of taking two pebbles off one vertex and placing one on an adjacent vertex. Theoptimal pebbling numberofG, denoted byπopt(G), is the smallest numbermsuch that for some distribution ofmpebbles onG, one pebble can be moved to any vertex ofGby a sequence of pebbling moves. LetPkbe the path onkvertices. Snevily defined then–kspindle graph as follows: takencopies ofPkand two extra verticesxandy, and then join the left endpoint (respectively, the right endpoint) of eachPktox(respectively,y), the resulting graph is denoted byS(n,k), and called then–kspindle graph. In this paper, we determine the optimal pebbling number for spindle graphs.
Highlights
Graph pebbling was rst introduced into the literature by Chung
The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves
The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that some distribution of m pebbles on G is solvable
Summary
Graph pebbling was rst introduced into the literature by Chung (see [1]). Pebbling has developed its own sub eld (see [2]). The optimal pebbling number of G, denoted by πopt(G), is the smallest number m such that for some distribution of m pebbles on G, one pebble can be moved to any vertex of G by a sequence of pebbling moves. We determine the optimal pebbling number for spindle graphs. Let D be a distribution of pebbles on the vertices of G, or a distribution on G.
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