Coloring by pushing vertices

Let \(G\) be a graph of order \(n\), maximum degree at most \(\Delta\), 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 \[\sigma : V(G) \to \mathbb{N}_{0} : u \mapsto (1+\rho(u)) d_G(u) + \sum\limits_{v\in N_G(u)} \rho(v),\] is a vertex coloring, that is, adjacent vertices receive different values under \(\sigma\). They show the existence of a proper pushing scheme \(\rho\) with \(\max\{\rho(u) : u \in V(G)\} \leq \Delta^2\) and conjecture that this upper bound can be improved to \(\Delta\). We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme \(\rho\) with \(\sum\limits_{u\in V(G)} \rho(u) \leq (2\Delta^2+\Delta)n/6\).