Finding the chromatic number of complete graph

The chromatic number of a graph is the minimum number of colors needed to color its vertices such that no two adjacent vertices share the same color. For a complete graph K_n, every vertex is connected to every other vertex, which makes it a special and straightforward case for graph coloring.

Problem Statement

What is the chromatic number of the complete graph K_n?

Solution

The following diagram shows a properly colored complete graph K₄, where each of the 4 vertices requires a different color −

In a complete graph K_n, each vertex is adjacent to all remaining (n − 1) vertices. Since every pair of vertices is connected by an edge, no two vertices can share the same color. Hence, each vertex requires its own unique color.

Therefore, the chromatic number of K_n is −

χ(K_n) = n

Examples for Small Complete Graphs

The following SVG shows the coloring for K₁ through K₄ −

The chromatic numbers for small complete graphs −

In every case, the chromatic number equals the number of vertices, confirming that χ(K_n) = n.

Why Fewer Colors Won't Work

Suppose we try to color K_n with fewer than n colors. Since every vertex is adjacent to every other vertex, if any two vertices share the same color, they would violate the coloring rule (adjacent vertices must have different colors). Therefore, it is impossible to use fewer than n colors, and n colors are always sufficient.

Conclusion

The chromatic number of a complete graph K_n is always equal to n, because every vertex is adjacent to every other vertex and no two vertices can share a color.