Triangular fractal approximating graphs and their covering paths and cycles

TL;DR

This paper studies two types of fractal approximating graphs related to the Sierpinski gasket, analyzing their structure, paths, cycles, and transformations, and deriving formulas for their properties.

Contribution

It introduces and analyzes two novel fractal graphs, providing recursive and explicit formulas for their cardinalities, paths, cycles, and transformations, revealing new integer sequences.

Findings

Derived formulas for vertices and edges of the graphs

Identified Hamiltonian and tiling paths and cycles

Discovered bijective transformations preserving properties

Abstract

We observe two kinds of fractal approximating graphs, the background structures of the generalized Sierpinski Arrowhead Curve independently of the recursive curves. Both graphs related to the generalized Sierpinski Gasket and based on a checked triangular generator pattern. In the Overall Graph we connect the corners of the up facing neighbouring dark tiles. In the Inscribed Graph we connect their centroids. We describe their cardinalities in general case with recursive and explicit formulas, the numbers of their vertices and edges, their edge covering Hamiltonian-paths and -cycles, and their tiling-paths and -cycles which cover all of their dark tiles. Some of these formations are unambiguously transformable into each other and these bijective pairs keep their basic properties after the transformation. Some of their cardinalities form new integer sequences.