Wheel graph

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In the mathematical discipline of graph theory, a wheel graph W_n is a graph with n vertices, formed by connecting a single vertex to all vertices of an (n-1)-cycle. The numerical notation for wheels is used inconsistently in the literature: some authors instead use n to refer to the length of the cycle, so that their W_n is the graph we denote W_n+1. A wheel graph can also be defined as the 1-skeleton of an (n-1)-gonal pyramid.

Wheel graphs are planar graphs, and as such have a unique planar embedding. 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.

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 (Buckley and Harary 1988).

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 cycle 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.

[edit] References

• Buckley, Fred; Harary, Frank (1988). "On the euclidean dimension of a wheel". Graphs and Combinatorics 4 (1): 23-30. DOI:10.1007/BF01864150. 

• 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. 

[edit] External links

• Eric W. Weisstein, Wheel Graph at MathWorld.