On the number of edges in a K5-minor-free graph of given girth

We consider the following question: what is the maximum number of edges in a K5-minor-free graph with n vertices and girth g, where n≥4? With no restriction on the girth (g=3), it is well known that the answer is 3n−6. For g=4 and n≥5, the answer is known to be 3n−9. For g=5, we show there are two graphs with 9n−205 edges and every other graph contains at most 9n−215 edges and equality holds for infinitely many graphs. For g=8 and large n the answer is 3n−92. For g=4k, k≥2, we conjecture the answer is 3k3k−2(n−3).