  The 3n+1 problem, or Collatz problem, is an extremely simple to state,
extremely hard to solve, problem. A number of Collatz graphs have been
presented to visualize the Collatz sequences. The Collatz graph is grown by
considering the bottom-up method with the inverse relation. If n is the Collatz
functional value of m, then n is connected by m. The concept is simple, the
tree-based graphs indeed provide a path starting from n down to the root, the
number of 1, for a given seed n, and demonstrate the generated Collatz
sequences eventually converges to 1. However, as a general case, due to the
irregular structures, no one has yet proved the completeness of the Collatz
graphs. By completeness we mean that the Collatz graph contains all positive
integers n. This paper proves the Collatz conjecture by constructing a Collatz
graph with the regular structure. The developed Collatz graph consists of
Collatz nodes located various levels of the graph. In the developed graph, each
node consists of all positive integers m which have the functional value n. A
set of simple, yet efficient connection rules is also developed to construct
the graph. Results show that the developed Collatz graph generates the Collatz
trajectories for all positive integers and the sequences converge to 1. This
proves the completeness of the developed Collatz graph and Collatz conjecture.
