An elementary proof of a tight $Δ+1$ bound for the additive chromatic number

Abstract

An additive coloring assigns positive integer labels to the vertices of a graph so that adjacent vertices have different sums of their neighbors' labels. The additive chromatic number $η(G)$ is the least $k$ for which labels from $[k]$ suffice. We give an elementary orientation argument proving that every finite simple graph $G$ satisfies $η(G) \leq Δ(G) + 1$. This argument yields a $(Δ(G) + 1)$-additive-coloring in polynomial time and improves upon the current best bound of $η(G) \leq Δ(G)^{2} - Δ(G) + 1$, valid for $Δ(G) \geq 2$, from Akbari, Ghanbari, Manaviyat, and Zare.