The Collatz conjecture and De Bruijn graphs

Abstract

We study variants of the well-known Collatz graph, by considering the action of the 3n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between these infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+1 function, we get similar relations with 2-adic and p-adic De Bruijn graphs.Comment: 9 pages, 8 figure

Circular slider graphs: de Bruijn, Kautz, Rauzy, lamplighters and spiders

“…• In spite of the enormous popularity of de Bruijn graphs and their various modifications, there have been very few attempts to extend this notion to infinite graphs. Laarhoven -de Weger [LdW13] in the course of a discussion of a link between de Bruijn graphs and the famous 3n + 1 Collatz conjecture (see Example 3.11) introduced the infinite p-adic de Bruijn graph with the vertex set A Z+ and the arrows w ∼ Sw, where p = |A|, and S : α 0 α 1 · · · → α 1 α 2 . .…”

The 3x+1 Problem For Rational Numbers : Invariance of Periodic Sequences in 3x+1 Problem

“…See [2], [8], where similar generalizations are used to prove some results related to undecidability properties. See [7], [12], where these generalizations are considered in the context of 2-adic numbers and q based numeral systems.…”

A Clustering Perspective of the Collatz Conjecture

“…Besides, some authors have devoted efforts to rewrite the conjecture in other terms (see, e.g., [5] for an approach in terms of algebraic and boolean fractals). Furthermore, some work has been developed concerning the representation and study of the Collatz conjecture in terms of graphs [6][7][8][9][10][11][12].…”