A Graph Theoretical Approach to the Collatz Problem

Abstract

Andrei et al. have shown in 2000 that the graph $\mathrm{C}$ of the Collatz function starting with root $8$ after the initial loop is an infinite binary tree $\mathbf{A}(8)$. According to their result they gave a reformulated version of the Collatz conjecture: the vertex set $V(\mathbf{A}(8))={\mathbb{Z}}^{+}$. In this paper an inverse Collatz function $\overrightarrow{C}$ with eliminated initial loop is used as generating function of a Collatz graph ${\mathrm{C}}_{\overrightarrow{C}}$. This graph can be considered as the union of one forest that stems from sequences of powers of 2 with odd start values and a second forest that is based on branch values $y=6k+4$ where two Collatz sequences meet. A proof that the graph ${\mathrm{C}}_{\overrightarrow{C}}(1)$ is an infinite binary tree ${\mathbf{A}}_{\overrightarrow{C}}$ with vertex set $V({\mathbf{A}}_{\overrightarrow{C}}(1))={\mathbb{Z}}^{+}$ completes the paper.

7 pages, 4 figures, converted to PDFLatex and text restructured, mathematical notations corrected, circuit proof replaced by stronger previous version Wrong value in figure 4 and the legend of figure 3 corrected