Coloring by pushing vertices

<p>Let <span class="math inline">\(G\)</span> be a graph of order <span class="math inline">\(n\)</span>, maximum degree at most <span class="math inline">\(\Delta\)</span>, and no component of order 2. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a <em>proper pushing scheme</em> of <span class="math inline">\(G\)</span> as a function <span class="math inline">\(\rho : V(G) \to \mathbb{N}_{0}\)</span> for which <span class="math display">\[\sigma : V(G) \to \mathbb{N}_{0} : u \mapsto (1+\rho(u)) d_G(u) + \sum\limits_{v\in N_G(u)} \rho(v),\]</span> is a vertex coloring, that is, adjacent vertices receive different values under <span class="math inline">\(\sigma\)</span>. They show the existence of a proper pushing scheme <span class="math inline">\(\rho\)</span> with <span class="math inline">\(\max\{\rho(u) : u \in V(G)\} \leq \Delta^2\)</span> and conjecture that this upper bound can be improved to <span class="math inline">\(\Delta\)</span>. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme <span class="math inline">\(\rho\)</span> with <span class="math inline">\(\sum\limits_{u\in V(G)} \rho(u) \leq (2\Delta^2+\Delta)n/6\)</span>.</p>