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On rational radii coin representations of the wheel graph

100%

Agnarsson G., Dunham J.

Discussiones Mathematicae - General Algebra and Applications

2013

tom 33

nr 2

167-199

A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the cosinus of the internal angles formed by the center coin and two consecutive petals of the flower. In that process we also derive a recursion that these irreducible polynomials satisfy.

Vertex coloring the square of outerplanar graphs of low degree

100%

Agnarsson G., Halldórsson M.

Discussiones Mathematicae Graph Theory

2010

tom 30

nr 4

619-636

Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree Δ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree Δ ≠ 6 is known. Hence, this mentioned chromatic number of 7 is the last and only unknown upper bound of the chromatic number in terms of Δ.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Discussiones Mathematicae Graph Theory

2 Agnarsson G.

1 Dunham J.

1 Halldórsson M.

1 2013

1 2010

Wyniki wyszukiwania

On rational radii coin representations of the wheel graph

Vertex coloring the square of outerplanar graphs of low degree