Labels like red and blue are only used when the number of colors is small, and normally it is understood that the labels are drawn from the integers. The terminology of using colors for vertex labels goes back to map coloring. a connection directly back to itself) could never be properly colored, it is understood that graphs in this context are loopless. When used without any qualification, a coloring of a graph is almost always a proper vertex coloring, namely a labeling of the graph’s vertices with colors such that no two vertices sharing the same edge have the same color. 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. That is partly for perspective, and partly because some problems are best studied in non-vertex form, as for instance is 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 the starting point of the subject, and other coloring problems can be transformed into a vertex version. For example, the following can be colored minimum 3 colors. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share 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.Ĭhromatic Number: The smallest number of colors needed to color a graph G is called its chromatic number. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share 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.