Diagrammatic analysis of the fourth Legendre transform

Analysis of the iterative solution of the equations of motion for the fourth Legendre transform shows that all graphs of the fourth transform are 4-irreducible except for an explicitly calculated four-reducible part. Knowledge of the graphs of the fourth transform is equivalent to solution of the combinatorial problem of resumming bare graphs into skeleton graphs (with respect to triple and quadruple vertices simultaneously). Explicit separation of the fourreducible part is equivalent to finding skeleton graphs responsible for overlapping effects in the equations for the triple and the quadruple vertex. The parquet approximation corresponds to neglect of all nontrivial 4-irreducible graphs of the fourth Legendre transform.