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The method of construction of the Golomb graph as a unit distance graph, by drawing an outer regular polygon connected to an inner twisted polygon or star polygon, has also been used for unit distance representations of the
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, the Golomb graph is a polyhedral graph
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-c ...
with 10 vertices and 18 edges
Edge or EDGE may refer to:
Technology Computing
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. It is named after Solomon W. Golomb
Solomon Wolf Golomb (; May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he inve ...
, who constructed it (with a non- planar embedding) as a unit distance graph
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these gr ...
that requires four colors in any graph coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...
. Thus, like the simpler Moser spindle, it provides a lower bound for the Hadwiger–Nelson problem
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color ...
: coloring the points of the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
so that each unit line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
has differently-colored endpoints requires at least four colors.
Construction
The method of construction of the Golomb graph as a unit distance graph, by drawing an outer regular polygon connected to an inner twisted polygon or star polygon, has also been used for unit distance representations of the Petersen graph
In the mathematics, mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertex (graph theory), vertices and 15 edge (graph theory), edges. It is a small graph that serves as a useful example and counterexample for ...
and of generalized Petersen graphs. As with the Moser spindle, the coordinates of the unit-distance embedding of the Golomb graph can be represented in the quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...