I assume knowledge of the Collatz conjecture. Here I'm looking only at total stopping times $t(n)$, and mostly will drop the word "total." Just looking at relatively simple graphs of stopping times on Wikipedia shows many patterns in stopping times, sets of overlapping discontinuous "curves". In ...

I will clarify the question by defining some straightforward concepts. Let G = (V, E) be a simple graph with chromatic number $k$. We call a subset $V'\subset V$ fully chromatic if every proper k-colouring of G uses also $k$ distinct colors on $V'$. We say such a fully chromatic set is minimal, i...