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.

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. Edges of unit distance graphs sometimes cross each other, so they are not always planar; a unit distance graph without crossings is called a matchstick graph.

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. Another important open problem concerning unit distance graphs asks how many edges they can have relative to their number of vertices.

Examples

The hypercube graph Q₄ as a unit distance graph.

The following graphs are unit distance graphs:

Any cycle graph
Any grid graph
Any hypercube graph
Any star graph
The Petersen graph
The Heawood graph (Gerbracht 2009)
The wheel graph W₇
The Moser spindle, the smallest 4-chromatic unit distance graph

Subgraphs of unit distance graphs

A unit distance drawing of the Möbius-Kantor graph in which some nonadjacent pairs are also at unit distance from each other.

Some sources define a graph as being a unit distance graph if its vertices can be mapped to distinct locations in the plane such that adjacent pairs are at unit distance apart, disregarding the possibility that some non-adjacent pairs might also be at unit distance apart. For instance, the Möbius-Kantor graph has a drawing of this type.

According to this looser definition of a unit distance graph, all generalized Petersen graphs are unit distance graphs (Žitnik, Horvat & Pisanski 2010). In order to distinguish the two definitions, the graphs in which non-edges are required to be a non-unit distance apart may be called strict unit distance graphs (Gervacio, Lim & Maehara 2008).

The graph formed by removing one of the spokes from the wheel graph W₇ is a subgraph of a unit distance graph, but is not a strict unit distance graph: there is only one way (up to congruence) to place the vertices at distinct locations such that adjacent vertices are a unit distance apart, and this placement also puts the two endpoints of the missing spoke at unit distance (Soifer 2008, p. 94).

Counting unit distances

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

Paul 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 $n\log n.$ By considering points in a square grid with carefully chosen spacing, Erdős found an improved lower bound of the form

$n^{{1+c/\log \log n}},$

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 Joel Spencer, Endre Szemerédi, and William Trotter (1984), is proportional to

$n^{{4/3}};$

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.

Representation of algebraic numbers and the Beckman–Quarles theorem

For every algebraic number A, it is possible to find a unit distance graph G in which some pair of vertices are at distance A in all unit distance representations of G (Maehara 1991, 1992). This result implies a finite version of the Beckman–Quarles theorem: for any two points p and q at distance A, there exists a finite rigid unit distance graph containing p and q such that any transformation of the plane that preserves the unit distances in this graph preserves the distance between p and q (Tyszka 2000). The full Beckman–Quarles theorem states that any transformation of the Euclidean plane (or a higher dimensional space) that preserves unit distances must be a congruence; that is, for the infinite unit distance graph whose vertices are all the points in the plane, any graph automorphism must be an isometry (Beckman & Quarles 1953).

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 & Rödl (1990) show that the dimension necessary to embed a graph in this way may be bounded by twice its maximum degree.

The dimension needed to embed a graph so that all edges have unit distance, and the dimension needed to embed a graph so that the edges are exactly the unit distance pairs, may greatly differ from each other: the 2n-vertex crown graph may be embedded in four dimensions so that all its edges have unit length, but requires at least n − 2 dimensions to be embedded so that the edges are the only unit-distance pairs (Erdős & Simonovits 1980).

Computational complexity

It is NP-hard, and more specifically complete for the existential theory of the reals, to test whether a given graph is a unit distance graph, or is a strict unit distance graph (Schaefer 2013).

References

Beckman, F. S.; Quarles, D. A., Jr. (1953), "On isometries of Euclidean spaces", Proceedings of the American Mathematical Society 4: 810–815, MR 0058193 .
Erdős, Paul (1946), "On sets of distances of n points", American Mathematical Monthly (The American Mathematical Monthly, Vol. 53, No. 5) 53 (5): 248–250, doi:10.2307/2305092, JSTOR 2305092 .
Erdős, Paul; Simonovits, Miklós (1980), "On the chromatic number of geometric graphs", Ars Combinatoria 9: 229–246 . As cited by Soifer (2008, p. 97).
Gerbracht, Eberhard H.-A. (2009), Eleven unit distance embeddings of the Heawood graph, arXiv:0912.5395 .
Gervacio, Severino V.; Lim, Yvette F.; Maehara, Hiroshi (2008), "Planar unit-distance graphs having planar unit-distance complement", Discrete Mathematics 308 (10): 1973–1984, doi:10.1016/j.disc.2007.04.050 .
Kuperberg, Greg (1992), The Erdos kitty: At least $9050 in prizes!, Posting to usenet groups rec.puzzles and sci.math .
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 (1992), "Extending a flexible unit-bar framework to a rigid one", Discrete Mathematics 108 (1-3): 167–174, doi:10.1016/0012-365X(92)90671-2, MR 1189840 .
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 .
Schaefer, Marcus (2013), "Realizability of graphs and linkages", in Pach, János, Thirty Essays on Geometric Graph Theory, Springer, pp. 461–482, doi:10.1007/978-1-4614-0110-0_24 .
Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, ISBN 978-0-387-74640-1 .
Spencer, Joel; Szemerédi, Endre; Trotter, William T. (1984), "Unit distances in the Euclidean plane", Graph Theory and Combinatorics, Academic Press, pp. 293–308 .
Tyszka, Apoloniusz (2000), "Discrete versions of the Beckman-Quarles theorem", Aequationes Mathematicae 59 (1-2): 124–133, doi:10.1007/PL00000119, MR 1741475 .
Žitnik, Arjana; Horvat, Boris; Pisanski, Tomaž (2010), All generalized Petersen graphs are unit-distance graphs, IMFM preprints 1109 .

External links

Venkatasubramanian, Suresh, "Problem 39: Distances among Point Sets in R² and R³", The Open Problems Project .
Weisstein, Eric W., "Unit-Distance Graph", MathWorld.

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