Can you assume this is a planar graph? If so, you have Euler's relation to play with.
Is the question the existence of a 4 or more cycle, or exactly a 4-cycle?
Let G be a simple graph with 28 edges and 10 vertices.
Show that G contains a 4-cycle.
My first idea is to observe that at least one vertex must have degree 3.
Then I examine three branches coming from degree 3 and I try to show that we find a cycle there (involve the degree at least 3 vertex or the branch). But I can't see how to do that.