Coloring by Pushing Vertices

TL;DR

The paper investigates pushing schemes that assign nonnegative pushes $ρ(u)$ to vertices so that the induced values $σ(u)= (1+ρ(u)) d_G(u) + \sum_{v∈N_G(u)} ρ(v)$ yield a proper vertex coloring, extending the 1-2-3-conjecture framework. It proves the conjectured bound $P^1(G)≤Δ$ for cubic graphs and for regular bipartite graphs, and shows the existence of pushing schemes with total push sum bounded by $(2Δ^2+Δ)n/6$. A greedy algorithm, combined with a probabilistic first-moment analysis, yields a general bound $P^t(G) ≤ \sum_u ρ(u) ≤ (n Δ (2Δ+1))/6$, with experimental evidence suggesting much tighter performance in practice. The results illuminate the structure of pushing schemes in regular and near-regular graphs and discuss potential refinements via girth-based arguments and broader generalizations.

Abstract

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 $\sum_{u\in V(G)}ρ(u)\leq \left(2Δ^2+Δ\right)n/6$.

Figures (1)

• Figure 1: The partition of $V(G)$, the values of $\rho$, and the possible edges.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['theorem0']}
• Proposition 3
• proof
• proof : Proof of Theorem \ref{['theorem1']}
• Proposition 4
• proof
• Proposition 5
• proof