Wheel Graph - Properties

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. Any 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 cycles in W_n (sequence A002061 in 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.

The chromatic polynomial of the wheel graph W_n is :

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.