A \emph{wheel} is a graph formed by a chordless cycle C and a vertex u not in C that has at least three neighbors in C. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a line graph or has a clique cutset. We prove that every planar graph that does not contain a wheel as an induced subgraph is 3-colorable.
@article{arxiv.1309.7120,
title = {Wheel-free planar graphs},
author = {Pierre Aboulker and Maria Chudnovsky and Paul Seymour and Nicolas Trotignon},
journal= {arXiv preprint arXiv:1309.7120},
year = {2015}
}