The diagonal-label characterization of complete Young graphs

Let a Young graph be the graph associated with the $(g,k)$-reverse multiples, whose node labels have the form $[r',r]$. A node is nonzero diagonal-labelled when its label is $[r,r]$ with $r\ne0$.

Complete Young graph conjecture. A Young graph is a complete graph if and only if it has at least one nonzero diagonal-labelled node $[r,r]$.

This is a stronger statement than the preceding theorem, which proves only that a Young graph is complete when every node label has the form $[r,r]$. The conjecture is consistent with all the data reported in the paper, but remains open.