A Note on Chromatic Sum

TL;DR

This paper explores properties of the chromatic sum of graphs, providing a necessary condition for graph homomorphisms and establishing an inequality relating the chromatic sum to the fractional chromatic number and the size of the graph.

Contribution

It introduces a necessary condition for the existence of graph homomorphisms and proves an inequality linking the chromatic sum with fractional chromatic number and graph order.

Findings

Established a necessary condition for graph homomorphisms.

Proved that the chromatic sum is less than the product of fractional chromatic number and graph size.

Abstract

The chromatic sum $\Sigma(G)$ 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 $\Sigma(G)<\chi_f(G)|G|$ for every graph $G$.