See the following exercise from Introduction to Graph Theory by Douglas West:
I managed to show part a) easily, mainly because it is a symmetric problem. I showed it as follows:
Without loss of generality, colour the middle vertex by 1. Then the $\bar 1$-vertex connected to the middle vertex has to be either 2 or 4. If it is 2, then the lower segment can’t be coloured. If it is 4, then the left segment can’t be coloured.
However, in exercise b), I don't even know what is a good place to start. And I'm wondering if I could use the result of a) for b)?
I tried systemizing/generalising the colouring, but I ended up with too many cases. Could someone point me in the right direction?
If the "extra" vertex is coloured $5$, then the left part is the graph from part a). If the "extra" vertex is coloured $4$, then the second part is the graph from part a). If the "extra" vertex is coloured $3$, then the right part is the graph from part a). If the "extra" vertex is coloured $2$, then the third part is the graph from part a).