graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, 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 Vertex (geometry), vertices and Edge (geometry), edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyh ...

with 10 vertices and 18

edges Enhanced Data rates for GSM Evolution (EDGE), also known as 2.75G and under various other names, is a 2G digital mobile phone technology for packet switched data transmission. It is a subset of General Packet Radio Service (GPRS) on the GS ...

. 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. He most notably inven ...

, who constructed it (with a non-

planar Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), ...

embedding) as a

unit distance graph In mathematics, particularly geometric graph theory, a unit distance graph is a Graph (discrete mathematics), graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly o ...

that requires four colors in any

graph coloring In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have th ...

. Thus, like the simpler

Moser spindle In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William, with seven vertices and eleven edges. It can be dra ...

, 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, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

so that each unit

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

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 mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph i ...

and of

generalized Petersen graph In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways o ...

s. 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 of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...

\mathbb

sqrt In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

/math>.

Fractional coloring

The

fractional chromatic number Fractional coloring is a topic in a branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices ...

of the Golomb graph is 10/3. The fact that this number is at least this large follows from the fact that the graph has 10 vertices, at most three of which can be in any independent set. The fact that the number is at most this large follows from the fact that one can find 10 three-vertex independent sets, such that each vertex is in exactly three of these sets. This fractional chromatic number is less than the number 7/2 for the Moser spindle and less than the fractional chromatic number of the unit distance graph of the plane, which is bounded between 3.6190 and 4.3599.

References

External links

*{{mathworld, title=Golomb Graph, id=GolombGraph Individual graphs Planar graphs