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	3 if n is odd 4 if n is even
Spectrum	$\{2 \cos(2 k \pi / (n-1))^1;$ $k = 1, \ldots, n - 2\}$$\cup \{1 \pm \sqrt{n}\}$
Properties	Hamiltonian Self-dual Planar
Notation	Wn

In the mathematical discipline of graph theory, a wheel graph is a graph formed by connecting a single 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 ≥ 4); other authors^[2] 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}}.^[3]

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 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.^[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.

References

1. Weisstein, Eric W., "Wheel Graph", MathWorld.
2. Rosen, Kenneth H. (2011). Discrete Mathematics and Its Applications, 7th ed., McGraw-Hill, p. 655.
3. Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 56. ISBN 978-0-486-67870-2. Retrieved 8 August 2012.
4. Buckley, Fred; Harary, Frank (1988), "On the euclidean dimension of a wheel", Graphs and Combinatorics 4 (1): 23–30, doi:10.1007/BF01864150.
5. Faudree, Ralph J.; McKay, Brendan D. (1993), "A conjecture of Erdős and the Ramsey number r(W₆)", J. Combinatorial Math. and Combinatorial Comput. 13: 23–31.