411 research outputs found
Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
Optimal Pebbling in Products of Graphs
We prove a generalization of Graham's Conjecture for optimal pebbling with
arbitrary sets of target distributions. We provide bounds on optimal pebbling
numbers of products of complete graphs and explicitly find optimal -pebbling
numbers for specific such products. We obtain bounds on optimal pebbling
numbers of powers of the cycle . Finally, we present explicit
distributions which provide asymptotic bounds on optimal pebbling numbers of
hypercubes.Comment: 28 pages, 1 figur
More on the -restricted optimal pebbling number
Let be a simple graph. A function is called a configuration of pebbles on the vertices of and the
weight of is which is just the total number of
pebbles assigned to vertices. A pebbling step from a vertex to one of its
neighbors reduces by two and increases by one. A pebbling
configuration is said to be solvable if for every vertex , there
exists a sequence (possibly empty) of pebbling moves that results in a pebble
on . A pebbling configuration is a -restricted pebbling configuration
(abbreviated RPC) if for all . The -restricted
optimal pebbling number is the minimum weight of a solvable RPC
on . Chellali et.al. [Discrete Appl. Math. 221 (2017) 46-53] characterized
connected graphs having small -restricted optimal pebbling numbers and
characterization of graphs with stated as an open problem.
In this paper, we solve this problem. We improve the upper bound of the
-restricted optimal pebbling number of trees of order . Also, we study
-restricted optimal pebbling number of some grid graphs, corona and
neighborhood corona of two specific graphs.Comment: 12 pages, 11 figure
The optimal pebbling number of staircase graphs
Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V
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