Coloring by Pushing Vertices

Let $G$ be a graph of order $n$, maximum degree at most $\mathrm{\Delta }$, 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 $\rho :V(G)\to {\mathbb{N}}_0$ for which $$\sigma :V(G)\to {\mathbb{N}}_0:u\mapsto (1+\rho (u))d_G(u)+\sum _{v\in N_G(u)}\rho (v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $\sigma$. They show the existence of a proper pushing scheme $\rho$ with $\max \{\rho (u):u\in V(G)\}\le {\mathrm{\Delta }}^2$ and conjecture that this upper bound can be improved to $\mathrm{\Delta }$. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme $\rho$ with ${\sum }_{u\in V(G)}\rho (u)\le (2{\mathrm{\Delta }}^2+\mathrm{\Delta })n\mathrm{/}6$.