A) Incorrect. Consider the following disconnected planar graph:
B) Incorrect. A graph is Eulearian if all vertices have even degree but a planar graph can have vertices with odd degree .
C) Correct. Any planar graph can be vertex colored with maximum 4 colors (4-color theorem) so that no two adjacent vertices have same color.
So consider a planar graph with n vertices having chromatic number 4. So atmost we have 4 set of vertices of each color having size n/4 (evenly distributed). If we remove a set, we still will have all the vertices which is atleast one of the endpoint of all the edges i.e. we have a vertex cover having size n-(n/4) = 3n/4.
D) Incorrect. Consider K4 graph. It has independent set size 1 which is less than 4/3.