38,455 research outputs found
The Radio Number of Grid Graphs
The radio number problem uses a graph-theoretical model to simulate optimal
frequency assignments on wireless networks. A radio labeling of a connected
graph is a function such that for every pair
of vertices , we have where denotes the diameter of and
the distance between vertices and . Let be the
difference between the greatest label and least label assigned to . Then,
the \textit{radio number} of a graph is defined as the minimum
value of over all radio labelings of . So far, there have
been few results on the radio number of the grid graph: In 2009 Calles and
Gomez gave an upper and lower bound for square grids, and in 2008 Flores and
Lewis were unable to completely determine the radio number of the ladder graph
(a 2 by grid). In this paper, we completely determine the radio number of
the grid graph for , characterizing three subcases of the
problem and providing a closed-form solution to each. These results have
implications in the optimization of radio frequency assignment in wireless
networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure
Enumerating planar locally finite Cayley graphs
We characterize the set of planar locally finite Cayley graphs, and give a
finite representation of these graphs by a special kind of finite state
automata called labeling schemes. As a result, we are able to enumerate and
describe all planar locally finite Cayley graphs of a given degree. This
analysis allows us to solve the problem of decision of the locally finite
planarity for a word-problem-decidable presentation.
Keywords: vertex-transitive, Cayley graph, planar graph, tiling, labeling
schemeComment: 19 pages, 6 PostScript figures, 12 embedded PsTricks figures. An
additional file (~ 438ko.) containing the figures in appendix might be found
at http://www.labri.fr/Perso/~renault/research/pages.ps.g
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