266 research outputs found
The Optimal Rubbling Number of Ladders, Prisms and M\"obius-ladders
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 and adjacent to a vertex , and an extra pebble is added at
vertex . A vertex is reachable from a pebble distribution if it is possible
to move a pebble to that vertex using rubbling moves. The optimal rubbling
number is the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable. We determine the optimal
rubbling number of ladders (), prisms () and
M\"oblus-ladders
Constructions for the optimal pebbling of grids
In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics]
the authors conjecture that if every vertex of an infinite square grid is
reachable from a pebble distribution, then the covering ratio of this
distribution is at most . First we present such a distribution with
covering ratio , disproving the conjecture. The authors in the above paper
also claim to prove that the covering ratio of any pebble distribution is at
most . The proof contains some errors. We present a few interesting
pebble distributions that this proof does not seem to cover and highlight some
other difficulties of this topic
Upper Bound on the Optimal Rubbling Number in graphs with given minimum degree
A pebbling move on a graph removes two pebbles at a vertex and adds
one pebble at an adjacent vertex. A vertex is reachable
from a pebble distribution if it is possible to move a pebble to
that vertex using pebbling moves. The optimal pebbling number is
the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable. Czygrinow proved that
the optimal pebbling number of a graph is at most , where is the number of the vertices and is
the minimum degree of the graph. We improve this result and show that the optimal pebbling number is at most
Solutions of Omitting Rail Expansion Joints in Case of Steel Railway Bridges with Wooden Sleepers
The Technical Specifications of D.12/H. of Hungarian State Railways (MĂV) specifies that a continuously welded rail (CWR) track can be constructed through a bridge without being interrupted if the expansion length of the bridge is not longer than 40 m. If the expansion length of a bridge is greater than 40 m, the continuously welded rail should normally be interrupted; a rail expansion joint has to be constructed. The goal of this research is to provide technical solutions of track structures on bridges so a continuously welded rail can be constructed through the bridge from an earthwork without interruption, so rail expansion joints can be omitted
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