Coloring Sierpiński graphs and Sierpiński gasket graphs

Abstract

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