2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75662We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controling the deformation theory of associative algebras and prove that there is a canonical solution within the class of graphs without circuits, without assuming the Jacobi identity. The proof is based on the unique factorization property of graph insertions.AMS LaTeX, 16 pagesQuantum AlgebraCombinatorics53D55; 14Fxxtext