2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217154The chromatic sum $Σ(G)$ of a graph $G$ is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present $Σ(G)<χ_f(G)|G|$ for every graph $G$.Combinatoricstext