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complete graph (Definition)

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:

\includegraphics[scale=1.0]{k4.eps}

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$.



"complete graph" is owned by vampyr.
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See Also: tournament

Other names:  complete, clique
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Cross-references: chromatic number, Hamiltonian path, odd, Euler circuit, degree, triangular number, number, adjacent, edges, contains, vertices
There are 40 references to this entry.

This is version 6 of complete graph, born on 2002-02-03, modified 2002-05-27.
Object id is 1757, canonical name is CompleteGraph.
Accessed 13367 times total.

Classification:
AMS MSC: 05C99 (Combinatorics :: Graph theory :: Miscellaneous)
Pending Errata and Addenda
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