complete graph

The complete graph with $n$ vertices, denoted $K_n$, contains all possible edges; that is, any two vertices are adjacent.

The complete graph of $4$ vertices, or $K_4$ looks like this:

The number of edges in $K_n$ is the $n-1$th triangular number. Every vertex in $K_n$ has degree $n-1$; therefore $K_n$ has an Euler circuit if and only if $n$ is odd. A complete graph always has a Hamiltonian path, and the chromatic number of $K_n$ is always $n$.