Majority additive coloring and the maximum degree
Christoph Brause, Dieter Rautenbach, Laurin Schwartze
TL;DR
This work investigates majority additive colorings $\chi_{mac}(G)$ of graphs, where a labeling $c:V(G)\to[k]$ must avoid any vertex $u$ (with $d_G(u)\ge2$) having more than half of its neighbors share the same neighbor-sum $s_c(v)=\sum_{w\in N_G(v)}c(w)$; graphs with no obstruction are called good. It establishes a quadratic universal bound $\chi_{mac}(G)\le 2\Delta(\Delta-1)+1$ for good graphs and a sublinear bound $\chi_{mac}(G)\le 4e^3\cdot \Delta^{4/\lfloor\delta/2\rfloor}$ under a private-neighbor condition, while showing there exist good graphs with $\chi_{mac}(G)\ge 2\Delta+1$ via Steiner triple system constructions; it also proves NP-completeness of deciding whether $\chi_{mac}(G)\le k$ for all $k\ge2$. The results disprove Kamyczura’s conjectures that $\chi_{mac}(G)$ is always small relative to $\Delta$, but demonstrate tightness of the growth bounds in general and under additional local conditions. The paper highlights the boundary between combinatorial constructions (lower bounds) and probabilistic/greedy methods (upper bounds) in the study of majority additive colorings.
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 $Δ$ admitting a majority additive coloring has a majority additive $\mathcal{O}\left(Δ^2\right)$-coloring. Under additional restrictions we improve this to sublinear in $Δ$. We show that determining whether a majority additive $k$-coloring exists for a given graph is NP-complete for all $k\geq 2$.