PLANAR GRAPHS WITH NO 6-WHEEL MINOR

Tutte's wheels theorem states that the k-spoked wheel graphs, W(k), ar e the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the gr aphs that do not have a minor isomorphic to W(k) for small values of k . Dirac determined that the graphs having no W3-minor are the series-p arallel networks. An easy consequence of Tutte's wheels theorem is tha t W3 is the only simple, 3-connected graph that has a W3-minor and no W4-minor. Oxley characterized the graphs that have a W4-minor and no W 5-minor. This paper characterizes the planar graphs that have a W5-min or and no W6-minor. A best-possible upper bound on the number of edges of such a graph is also determined.