Graphs \(S[n,k]\) are introduced as the graphs obtained from the Sierpiński graphs \(S(n, k)\) by contracting edges that lie in no complete subgraph \(K_k\). The family \(S[n,k]\) generalizes the previously studied class of Sierpiński gasket graphs \(S_k\). We investigate various properties of graphs \(S[n,k]\), particularly focusing on hamiltonicity and chromatic number.