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 graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs. A problem posed by Paul Erdős known as the unit distance problem asks for the maximum possible number of unit-distance pairs determined by n {\displaystyle n} points in the Euclidean plane; equivalently, it asks for the maximum number of edges in a unit distance graph on n {\displaystyle n} vertices. The best known upper bound is O ( n 4 / 3 ) . {\displaystyle O(n^{4/3}).} The best known lower bound is Ω ( n 1.014 ) {\displaystyle \Omega (n^{1.014})} (for sufficiently large n ) {\displaystyle n)} . The number of colors required to color T unit distance graphs is also unknown; in the equivalent Hadwiger–Nelson problem for the plane, some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number α , {\displaystyle \alpha ,} there is a unit distance graph with two vertices that must be at distance α . {\displaystyle \alpha .} According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries. It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard, and more specifically complete for the existential theory of the reals.
Definition
The unit distance graph for a set of points in the plane is the undirected graph having those points as its vertices, with an edge between two vertices whenever their Euclidean distance is exactly one. An abstract graph is said to be a unit distance graph if it is possible to find distinct locations in the plane for its vertices, so that its edges have unit length and so that all non-adjacent pairs of vertices have non-unit distances. When this is possible, the abstract graph is isomorphic to the unit distance graph of the chosen locations. Alternatively, some sources use a broader definition, allowing non-adjacent pairs of vertices to be at unit distance. The resulting graphs are the subgraphs of the unit distance graphs (as defined here). Where the terminology may be ambiguous, the graphs in which non-edges must be a non-unit distance apart may be called strict unit distance graphs or faithful unit distance graphs. The subgraphs of unit distance graphs are equivalently the graphs that can be drawn in the plane using only one edge length. For brevity, this article refers to these as "non-strict unit distance graphs". Unit distance graphs should not be confused with unit disk graphs, which connect pairs of points when their distance is less than or equal to one, and are frequently used to model wireless communication networks.
Examples The complete graph on two vertices is a unit distance graph, as is the complete graph on three vertices (the triangle graph), but not the complete graph on four vertices. Generalizing the triangle graph, every cycle graph is a unit distance graph, realized by a regular polygon. Two finite unit distance graphs, connected at a single shared vertex, yield another unit distance graph, as one can be rotated with respect to the other to avoid undesired additional unit distances. By thus connecting graphs, every finite tree or cactus graph may be realized as a unit distance graph.
Any Cartesian product of unit distance graphs produces another unit distance graph; however, the same is not true for some other common graph products. For instance, the strong product of graphs, applied to any two non-empty graphs, produces complete subgraphs with four vertices, which are not unit distance graphs. The Cartesian products of path graphs form grid graphs of any dimension, the Cartesian products of the complete graph on two vertices are the hypercube graphs, and the Cartesian products of triangle graphs are the Hamming graphs H ( d , 3 ) {\displaystyle H(d,3)} . Other specific graphs that are unit distance graphs include the Petersen graph, the Heawood graph, the wheel graph W 7 {\displaystyle W_{7}} (the only wheel graph that is a unit distance graph), and the Moser spindle and Golomb graph (small 4-chromatic unit distance graphs). All generalized Petersen graphs, such as the Möbius–Kantor graph depicted, are non-strict unit distance graphs. Matchstick graphs are a special case of unit distance graphs, in which no edges cross. Every matchstick graph is a planar graph, but some otherwise-planar unit distance graphs (such as the Moser spindle) have a crossing in every representation as a unit distance graph. Additionally, in the context of unit distance graphs, the term 'planar' should be used with care, as some authors use it to refer to the plane in which the unit distances are defined, rather than to a prohibition on crossings. The penny graphs are an even more special case of unit distance and matchstick graphs, in which every non-adjacent pair of vertices are more than one unit apart.
Properties
Number of edges
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