132 research outputs found
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Color-blind index in graphs of very low degree
Let be an edge-coloring of a graph , not necessarily
proper. For each vertex , let , where is
the number of edges incident to with color . Reorder for
every in in nonincreasing order to obtain , the color-blind
partition of . When induces a proper vertex coloring, that is,
for every edge in , we say that is color-blind
distinguishing. The minimum for which there exists a color-blind
distinguishing edge coloring is the color-blind index of ,
denoted . We demonstrate that determining the
color-blind index is more subtle than previously thought. In particular,
determining if is NP-complete. We also connect
the color-blind index of a regular bipartite graph to 2-colorable regular
hypergraphs and characterize when is finite for a class
of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi
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