Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. Note: Many terms used in this article are defined in Glossary of graph theory. Graph coloring is still a very active field of research. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. Graph coloring enjoys many practical applications as well as theoretical challenges. The nature of the coloring problem depends on the number of colors but not on what they are. In general, one can use any finite set as the 'color set'. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the 'colors'. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. This was generalized to coloring the faces of a graph embedded in the plane. The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This is partly pedagogical, and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring. However, non-vertex coloring problems are often stated and studied as-is. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. Vertex coloring is usually used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. Matsui, Tomomi (2000), 'Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs', Lecture Notes in Computer Science, Lecture Notes in Computer Science, 1763: 194–200, doi: 10.1007/978-5-7_16, ISBN 978-1-7.(1994), Geometry based heuristics for unit disk graphs, arXiv: math.CO/9409226. Müller, Tobias (2011), 'Sphere and dot product representations of graphs', Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SoCG'11), June 13–15, 2011, Paris, France, pp. 308–314. An example of a graph that is not a unit disk graph is the star K 1, 6 -free graphs', Discrete Applied Mathematics, 284: 53–60, doi: 10.1016/j.dam.2020.03.024, MR 4115456 Unit disk graphs may be formed in a different way from a collection of equal-radius circles, by connecting two circles with an edge whenever one circle contains the center of the other circle.Įvery induced subgraph of a unit disk graph is also a unit disk graph.These graphs have a vertex for each circle or disk, and an edge connecting each pair of circles or disks that have a nonempty intersection. Unit disk graphs are the intersection graphs of equal-radius circles, or of equal-radius disks.Unit disk graphs are the graph formed from a collection of points in the Euclidean plane, with a vertex for each point and an edge connecting each pair of points whose distance is below a fixed threshold.There are several possible definitions of the unit disk graph, equivalent to each other up to a choice of scale factor:

Unit disk graph chromatic bound