Wheel graph

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

In the mathematical discipline of 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 write W_n to denote a wheel graph with n vertices (n ≥ 4); other authors instead use W_n to denote a wheel graph with n+1 vertices (n ≥ 3), which is formed by connecting a single vertex to all vertices of a cycle of length n. In the rest of this article we use the former notation.

Set-builder construction

Given a vertex set of {1, 2, 3, …, v}, the edge set of the wheel graph can be represented in set-builder notation by {{1, 2}, {1, 3}, …, {1, v}, {2, 3}, {3, 4}, …, {v − 1, v}, {v, 2}}.

Properties

Wheel graphs are planar graphs, and as such 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).

The 7 cycles of the wheel graph W₄.

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.

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. 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.