COLORING SIERPIN´ SKI GRAPHS AND SIERPIN´ SKI GASKET GRAPHS

Abstract

Sierpiński graphs $S(n,3)$ are the graphs of the Tower of Hanoi puzzle with $n$ disks, while Sierpiński gasket graphs $S_n$ are the graphs naturally defined by the finite number of iterations that lead to the Sierpiński gasket. An explicit labeling of the vertices of $S_n$ is introduced. It is proved that $S_n$ is uniquely 3-colorable, that $S(n,3)$ is uniquely 3-edge-colorable, and that $\chi'(S_n)=4$, thus answering a question from~[15]. It is also shown that $S_n$ contains a 1-perfect code only for $n=1$ or $n=3$ and that every $S(n,3)$ contains a unique Hamiltonian cycle.