Proper Coloring
In graph theory, a proper coloring of a graph is an assignment of colors to the vertices of the graph such that no adjacent vertices have the same color
In graph theory, a proper coloring of a graph is an assignment of colors to the vertices of the graph such that no adjacent vertices have the same color. The colors used in a proper coloring are typically represented by integers, where each integer represents a different color.
More formally, let G = (V, E) be a graph with vertex set V and edge set E. A proper coloring of G is a function c: V -> C, where c(v) assigns a color from a set of colors C to each vertex v in V, such that for every edge (u, v) in E, c(u) ≠ c(v) (vertices u and v have different colors).
The minimum number of colors required for a proper coloring of a graph G is known as its chromatic number. The chromatic number is denoted as χ(G). Finding the chromatic number of a graph is an important problem in graph theory, and it can often be quite challenging.
Proper colorings have various applications, including in scheduling problems, map coloring, and register allocation in computer science. They also provide insights into the nature of graphs and their connectivity properties.
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