[Paper Review] Sierpi\' nski Gasket Graphs and Some of Their Properties

Alberto Teguia, Anant P. Godbole|arXiv (Cornell University)|Sep 12, 2005

Mathematical Dynamics and Fractals12 references19 citations

TL;DR

This paper studies the finite Sierpiński gasket graph $S_n$, derived from the first $n$ iterations of the Sierpiński fractal construction. It establishes that $S_n$ is Hamiltonian and pancyclic, determines its domination number (asymptotically 90% efficient), and derives a recursive formula for its cover pebbling number, revealing non-trivial computation despite the stacking theorem. The key contribution is a precise recursion for the cover pebbling number of $S_n$, highlighting structural complexity in this fractal-like graph.

ABSTRACT

The {\it Sierpiński fractal} or {\it Sierpiński gasket} $Σ$ is a familiar object studied by specialists in dynamical systems and probability. In this paper, we consider a graph $S_n$ derived from the first $n$ iterations of the process that leads to $Σ$, and study some of its properties, including its cycle structure, domination number and pebbling number. Various open questions are posed.

Motivation & Objective

• To analyze the structural and combinatorial properties of the finite Sierpiński gasket graph $S_n$, derived from the first $n$ iterations of the Sierpiński fractal.
• To determine the domination number of $S_n$ and show its asymptotic efficiency is 90%, contrasting with 100% efficiency in other Sierpiński graphs.
• To compute the cover pebbling number $\lambda(S_n)$, showing it is non-trivial despite the stacking theorem, and derive a recursive formula.
• To investigate cycle structure, chromatic number, and pebbling properties of $S_n$, extending known results on fractal graphs.

Proposed method

• Define $S_n$ as the graph formed by the vertices and edges of the $n$-th iteration of the Sierpiński gasket construction, with $|V_n| = \frac{3}{2}(3^{n-1} + 1)$ and $|E_n| = 3^n$.
• Use induction and recursive decomposition: $S_{n+1}$ consists of three copies of $S_n$ (top, bottom-left, bottom-right), connected at vertices of degree two.
• Prove Hamiltonicity and pancyclicity by constructing Hamiltonian paths between degree-two vertices and combining them across components.
• Establish 3-colorability via inductive vertex coloring, assigning colors different from adjacent vertices in the base and recursive steps.
• Apply the stacking theorem of Vuong and Wyckoff to compute $\lambda(S_n)$, showing the worst-case vertex for pebbling is a corner vertex of degree two.
• Derive a recursion for $\lambda(S_{n+1})$ using distance distribution from a corner vertex, based on $\beta(i)$, the number of vertices at distance $i$ from a corner.

Experimental results

Research questions

• RQ1 Is the Sierpiński gasket graph $S_n$ Hamiltonian and pancyclic for all $n \geq 1$?
• RQ2 What is the domination number of $S_n$, and what is its asymptotic efficiency?
• RQ3 What is the cover pebbling number $\lambda(S_n)$, and can it be computed via a non-trivial recursion?
• RQ4 What are the chromatic index and total chromatic number of $S_n$?
• RQ5 How do structural properties like cycle structure and domination number generalize to Sierpiński-like graphs derived from Pascal’s triangle modulo $p$?

Key findings

• The graph $S_n$ has $\frac{3}{2}(3^{n-1} + 1)$ vertices and $3^n$ edges, derived from recursive construction via adding downward triangles.
• $S_n$ is properly 3-colorable, with $\chi(S_n) = 3$, proven by inductive vertex coloring preserving proper adjacency.
• $S_n$ is Hamiltonian and pancyclic, meaning it contains cycles of all lengths from 3 to $|V_n|$, established via recursive path construction.
• The domination number of $S_n$ is asymptotically 90% efficient, meaning the domination number grows as $\sim \frac{1}{10} |V_n|$, contrasting with 100% efficiency in other Sierpiński graphs.
• The cover pebbling number $\lambda(S_n)$ satisfies the recursion $\lambda(S_{n+1}) = (1 + 2^{2^{n-1}+1})\lambda(S_n) - (2^{2^n} + 2^{2^{n-1}+1})$, derived from distance distribution and the stacking theorem.
• The diameter of $S_n$ is $2^{n-1}$, and the worst-case vertex for pebbling (maximizing $ST(v) = \sum_u 2^{d(u,v)}$) is a corner vertex of degree two, proven by induction on component distances.