Majority additive coloring and the maximum degree

Abstract

Kamyczura introduced the notion of a majority additive $k$-coloring of a graph $G$ as a function $c: V(G) \to \{1,2,\ldots,k\}$ such that $$\left|\left\{u \in N_G(v):\sum_{w \in N_G(u)} c(w) = s \right\}\right|\leq \max\left\{1,\frac{d_G(v)}{2}\right\}$$ for every vertex $v$ of $G$ and every positive integer $s$. We show that every graph $G$ of maximum degree $\Delta$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(\Delta^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $\Delta$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\geq 2$.