Are you working back and forth? You’ll run into trouble if you try to knit it in the round.
There is a thread on intarsia in the round recently, so if you want to check that out, there are some hints.
Otherwise, if you’re working back and forth, you’ll work with your main color up to the star, then knit with your contrasting color, and start a new ball of main color for the other side of the star. For the first stitch, you’ll have a tail to weave in, but after that just hold the color you’re no longer using over to the left and bring the new color up from under it. You’ll avoid holes.
www.knittingpatterncentral.com has a good link to intarsia knitting in it’s instruction section.