I've run into an exercise while studying Modern Graph Theory by Bela Bollobas. I'm trying to solve it. I probably need to use discharging method. However, I couldn't simply grasp the idea of applying it. Can anybody help please? Thanks in advance.
Let G be a maximal planar graph of order at least 25 and minimal degree 5. Call a vertex a major vertex if its degree is at least 7 otherwise" call it minor. Then G contains one of the following:
- a minor vertex with 3 consecutive neighbours of degree 5,
- a vertex of degree 5 with minor neighbours only)
- a major vertex with at most one neighbour of degree at least 6.