A Graph Theoretical Approach to the Collatz Problem

Abstract

Andrei et al. have shown in 2000 that the graph $\boldsymbol{\mathrm{C}}$ of the Collatz function starting with root $8$ after the initial loop is an infinite binary tree $\boldsymbol{A}(8)$. According to their result they gave a reformulated version of the Collatz conjecture: the vertex set $V(\boldsymbol{A}(8))=\mathbb{Z}^+$. In this paper an inverse Collatz function $\overrightarrow{C}$ with eliminated initial loop is used as generating function of a Collatz graph ${\boldsymbol{\mathrm{C}}}_{\overrightarrow{C}}$. 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 ${\boldsymbol{\mathrm{C}}}_{\overrightarrow{C}}(1)$ is an infinite binary tree $\boldsymbol{A}_{\overrightarrow{C}}$ with vertex set $V({\boldsymbol{A}}_{\overrightarrow{C}}(1))=\mathbb{Z}^+$ completes the paper.