counting the boundary and internal edges of a connected component of hypercubes

If we obtain two values which represent the number of vertices and edges in hypercubes respectively, do we can determine the size of all possible connected components? For example, there are totlal 16 vertices and 32 edges in a 4-dimensional hypercube. Now we assume 10 vertices and 13 edges are remained in this hypercube. What are all possible cases?





Dose it exist two or more connected components in it? In fact, there is at most "one" connected component by giving such number of vertices and edges. But the problem is coming again, how to prove it? Oh! my head becomes bigger and bigger again...













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