Connectivity and some other properties of generalized Sierpiński graphs

If G is a graph and n a positive integer, then the generalized Sierpiński graph S G is a fractal-like graph that uses G as a building block. The construction of S G generalizes the classical Sierpiński graphs S n p , where the role of G is played by the complete graph Kp. An explicit formula for the number of connected components in S G is given and it is proved that the (edge-)connectivity of S G equals the (edge-)connectivity of G. It is demonstrated that S n G contains a 1-factor if and only if G contains a 1-factor. Hamiltonicity of generalized Sierpiński graphs is also discussed.