Unit distance graph

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The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length.
The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length.
The hypercube graph Q4 as a unit distance graph.
The hypercube graph Q4 as a unit distance graph.

In mathematics, and 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 by an edge whenever the distance between the two points is exactly one. Unlike planar graphs, edges of unit distance graphs are allowed to cross each other.

The Hadwiger–Nelson problem concerns the chromatic number of unit distance graphs. It is known that there exist unit distance graphs requiring four colors in any proper coloring, and that all such graphs can be colored with at most seven colors.

For any algebraic number A, it is possible to find a unit distance graph G in which some pair of vertices are at distance A in any unit distance representation of G (Maehara 1991).

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[edit] Examples

The following graphs are unit distance graphs:

[edit] Counting unit distances

Unsolved problems in mathematics: How many unit distances can be determined by a set of n points?

Erdős (1946) posed the problem of estimating how many pairs of points in a set of n points could be at unit distance from each other. In graph theoretic terms, how dense can a unit distance graph be?

The hypercube graph provides a lower bound on the number of unit distances proportional to nlogn. By considering points in a square grid with carefully chosen spacing, Erdős found an improved lower bound of the form

n^{1+\frac{c}{\log\log n}},

and offered a prize of $500 for determining whether or not the maximum number of unit distances can also be upper bounded by a function of this form (Kuperberg 1992). The best known upper bound for this problem, due to Spencer, Szemerédi, and Trotter (1984), is proportional to

n^{\frac43};

this bound can also be viewed as counting incidences between points and unit circles, and is closely related to the Szemerédi–Trotter theorem on incidences between points and lines.

[edit] Generalization to higher dimensions

The definition of a unit distance graph may naturally be generalized to any higher dimensional Euclidean space. Any graph may be embedded as a set of points in a sufficiently high dimension; Maehara and Rödl (1990) show that the dimension necessary to embed a graph in this way may be bounded by twice its maximum degree.

[edit] See also

[edit] References

  • Maehara, Hiroshi (1991). "Distances in a rigid unit-distance graph in the plane". Discrete Applied Mathematics 31 (2): 193–200. doi:10.1016/0166-218X(91)90070-D. 
  • Maehara, Hiroshi; Rödl, Vojtech (1990). "On the dimension to represent a graph by a unit distance graph". Graphs and Combinatorics 6 (4): 365–367. doi:10.1007/BF01787703. 
  • Spencer, Joel; Szemerédi, Endre; Trotter, William T. (1984). "Unit distances in the Euclidean plane". Graph Theory and Combinatorics: 293–308, Academic Press. 

[edit] External links