Wheel graph

Vertices

n \geq 4

Edges

2(n - 1)

Diameter2 if

n > 4

1 if

n = 4

Girth3

Wheel graph
Several examples of wheel graphs
Vertices	$n \geq 4$
Edges	$2(n - 1)$
Diameter	2 if $n > 4$ 1 if $n = 4$
Girth	3
Chromatic number	3 if $n$ is odd 4 if $n$ is even
Spectrum	$\{2\cos(2k\pi /(n-1))^{1};$ $k=1,\ldots ,n-2\}$ $\cup \{1\pm {\sqrt {n}}\}$
Properties	Hamiltonian Self-dual Planar
Notation	$W n$
Table of graphs and parameters

In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with $n$ vertices can also be defined as the 1-skeleton of an ( $n - 1$ )-gonal pyramid.

Some authors^[1] write $W n$ to denote a wheel graph with $n$ vertices ( $n \geq 4$ ); other authors^[2] instead use $W n$ to denote a wheel graph with $n + 1$ vertices ( $n \geq 3$ ), which is formed by connecting a single vertex to all vertices of a cycle of length $n$ . The former notation is used in the rest of this article and in the table on the right.

Properties

Wheel graphs are planar graphs, and have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than K₄ = W₄, contains as a subgraph either W₅ or W₆.

There is always a Hamiltonian cycle in the wheel graph and there are $n^{2}-3n+3$ cycles in W_n (sequence A002061 in the OEIS).

For odd values of n, W_n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W_n has chromatic number 4, and (when n ≥ 6) is not perfect. W₇ is the only wheel graph that is a unit distance graph in the Euclidean plane.^[4]

The chromatic polynomial of the wheel graph W_n is :

P_{W_{n}}(x)=x((x-2)^{(n-1)}-(-1)^{n}(x-2)).

In matroid theory, two particularly important special classes of matroids are the wheel matroids and the whirl matroids, both derived from wheel graphs. The k-wheel matroid is the graphic matroid of a wheel W_k+1, while the k-whirl matroid is derived from the k-wheel by considering the outer cycle of the wheel, as well as all of its spanning trees, to be independent.

The wheel W₆ supplied a counterexample to a conjecture of Paul Erdős on Ramsey theory: he had conjectured that the complete graph has the smallest Ramsey number among all graphs with the same chromatic number, but Faudree and McKay (1993) showed W₆ has Ramsey number 17 while the complete graph with the same chromatic number, K₄, has Ramsey number 18.^[5] That is, for every 17-vertex graph G, either G or its complement contains W₆ as a subgraph, while neither the 17-vertex Paley graph nor its complement contains a copy of K₄.

Properties

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