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. See also: History of the four color theorem and History of graph theory 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 often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color this is called a vertex coloring. In graph theory, graph coloring is a special case of graph labeling it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. Ffi*, a APr*offi2on rttrift: W*c.A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible.
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