A Note on Chromatic Sum

Abstract

The chromatic sum \(\Sigma(G)\) of a graph \(G\) is the smallest sum of colors among all proper colorings using natural numbers. In this paper, we establish a necessary condition for the existence of graph homomorphisms. Furthermore, we show that \(\Sigma(G) \leq \chi_f(G) |V(G)|\) holds for every graph \(G\).