Laurin Schwartze

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 $k\geq 2$.

Let $G$ be a graph of order $n$, maximum degree at most $Δ$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function $ρ:V(G)\to\mathbb{N}_0$ for which $$σ:V(G)\to\mathbb{N}_0:u\mapsto \left(1+ρ(u)\right)d_G(u)+\sum_{v\in N_G(u)}ρ(v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $σ$. They show the existence of a proper pushing scheme $ρ$ with $\max\{ ρ(u):u\in V(G)\}\leq Δ^2$ and conjecture that this upper bound can be improved to $Δ$. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme $ρ$ with $\sum_{u\in V(G)}ρ(u)\leq \left(2Δ^2+Δ\right)n/6$.