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 $c:V(G)\to \{1,2,\dots ,k\}$ such that $$\left|\left\{u\in N_G(v):\sum _{w\in N_G(u)}c(w)=s\right\}\right|\le \max \left\{1,\frac{d_G(v)}{2}\right\}$$ for every vertex $v$ of $G$ and every positive integer $s$. We show that every graph $G$ of maximum degree $\hat{I}”$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(\hat{I}{”}^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $\hat{I}”$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\ge 2$.