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Coloring Some Finite Sets in ℝn

100%

Balogh J., Kostochka A., Raigorodskii A.

Discussiones Mathematicae Graph Theory

2013

tom 33

nr 1

25-31

This note relates to bounds on the chromatic number χ(ℝn) of the Euclidean space, which is the minimum number of colors needed to color all the points in ℝn so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in ℝn was introduced showing that . For many years, this bound has been remaining the best known bound for the chromatic numbers of some lowdimensional spaces. Here we prove that and find an exact formula for the chromatic number in the case of n = 2k and n = 2k − 1.

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Orientation distance graphs revisited

100%

Goddard W., Kanakadandi K.

Discussiones Mathematicae Graph Theory

2007

tom 27

nr 1

125-136

The orientation distance graph 𝓓ₒ(G) of a graph G is defined as the graph whose vertex set is the pair-wise non-isomorphic orientations of G, and two orientations are adjacent iff the reversal of one edge in one orientation produces the other. Orientation distance graphs was introduced by Chartrand et al. in 2001. We provide new results about orientation distance graphs and simpler proofs to existing results, especially with regards to the bipartiteness of orientation distance graphs and the representation of orientation distance graphs using hypercubes. We provide results concerning the orientation distance graphs of paths, cycles and other common graphs.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

1 Balogh J.

1 Goddard W.

1 Kanakadandi K.

1 Kostochka A.

1 Raigorodskii A.

1 2013

1 2007

Wyniki wyszukiwania

Coloring Some Finite Sets in ℝn

Orientation distance graphs revisited