graph is planar so we can directly say it requires not more than 4 color

this graph has also odd length cycle for that 3 color requires

also this graph has even length cycle for that 2 color

but i have a doubt here :

if graph is very complex and if both the even and odd length cycle appears as well as planar solution then whts the correct aproach to find color requirement in graph

same situation with this https://gateoverflow.in/3263/gate2008-it-3

@Bikram sir which approach ?pls help