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A unit distance graph with 16 vertices and 40 edges. 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 ...
Kalai linked additional posts by Jordan Ellenberg and Noam Elkies, with Elkies and (separately) de Grey proposing a Polymath project to find non-4-colorable unit distance graphs with fewer vertices than the one in de Grey's construction. [6] As of 2021, the smallest known unit distance graph with chromatic number 5 has 509 vertices. [7]
The induced subgraphs of this graph are called unit distance graphs. A seven-vertex unit distance graph, the Moser spindle, requires four colors; in 2018, much larger unit distance graphs were found that require five colors. The whole infinite graph has a known coloring with seven colors based on a hexagonal tiling of the plane.
The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length. The Petersen graph can also be drawn (with crossings) in the plane in such a way that all the edges have equal length. That is, it is a unit distance graph.
The smallest 3-regular matchstick graph is formed from two copies of the diamond graph placed in such a way that corresponding vertices are at unit distance from each other; its bipartite double cover is the 8-crossed prism graph. [2] In 1986, Heiko Harborth presented the graph that became known as the Harborth Graph.
Laman graph. Table of graphs and parameters. 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, [ 1] with seven vertices and eleven edges. It is a unit distance graph requiring four colors in any graph ...
The Cartesian product of two edges is a cycle on four vertices: K 2 K 2 = C 4. The Cartesian product of K 2 and a path graph is a ladder graph. The Cartesian product of two path graphs is a grid graph. ( K 2 ) n = Q n . {\displaystyle (K_ {2})^ {\square n}=Q_ {n}.} Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q ...
In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other. They are commonly formed from a Poisson ...