In star graph with N nodes has two category of nodes
one category with degree 1 and the other category with degree N-1
First category will contain N-1 nodes ans second category contains one node only
Example,
So we can divide the star graph into two category of nodes and nodes in same category doesn't has an an edge between them.The first category can be colored with single colour and the second category can be coloured with another colour.
There for we can call a star graph as bipitrate graph and hence we only need two colors for colouring a star graph.