Coloring by Pushing Vertices

TL;DR

This paper investigates a vertex coloring method called proper pushing schemes, proving the conjecture for certain graph classes and establishing bounds on the scheme's parameters.

Contribution

The paper proves the conjecture for cubic and regular bipartite graphs and provides bounds on the total pushing scheme values for general graphs.

Findings

Conjecture holds for cubic graphs.

Conjecture holds for regular bipartite graphs.

Bounds on total pushing scheme values are established.

Abstract

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 $$\sigma:V(G)\to\mathbb{N}_0:u\mapsto \left(1+\rho(u)\right)d_G(u)+\sum_{v\in N_G(u)}\rho(v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $\sigma$. They show the existence of a proper pushing scheme $\rho$ with $\max\{ \rho(u):u\in V(G)\}\leq \Delta^2$ 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 $\sum_{u\in V(G)}\rho(u)\leq \left(2\Delta^2+\Delta\right)n/6$.