منابع مشابه
On -unit distance graphs
We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the -unit distance graph to be the graph whose vertices are the points of E, in which two points are adjacent whenever their distance is within of 1. For certain values of we are able to show that the chromatic number is exactly seven. For some smaller values we show the chromatic number...
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A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particula...
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First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...
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Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0, a, b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families...
متن کاملDistance preserving graphs and graph products
If G is a graph then a subgraph H is isometric if, for every pair of vertices u, v of H, we have dH(u, v) = dG(u, v) where d is the distance function. We say a graph G is distance preserving (dp) if it has an isometric subgraph of every possible order up to the order of G. We give a necessary and sufficient condition for the lexicographic product of two graphs to be a dp graph. A graph G is seq...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2010
ISSN: 0012-365X
DOI: 10.1016/j.disc.2009.11.035