标题： 通过推动顶点着色 显示英文标题

标题： Coloring by Pushing Vertices

摘要： 设 $G$ 为一个阶数为 $n$、最大度数至多为 $\Delta$ 且没有阶数为 $2$ 的连通分支的图。 受著名的1-2-3猜想的启发，Bensmail、Marcille和Orenga定义了$G$的一种合适的推动方案为函数$\rho:V(G)\to\mathbb{N}_0$，其中$$\sigma:V(G)\to\mathbb{N}_0:u\mapsto \left(1+\rho(u)\right)d_G(u)+\sum_{v\in N_G(u)}\rho(v)$$是一个顶点着色，即相邻顶点在$\sigma$下接收不同的值。他们证明了存在一个合适的推动方案$\rho$，且$\max\{ \rho(u):u\in V(G)\}\leq \Delta^2$，并猜测这个上界可以改进为$\Delta$。我们证明了该猜想对于三次图和正则二分图成立。 此外，我们证明了存在一个合适的推进方案 $\rho$ 和 $\sum_{u\in V(G)}\rho(u)\leq \left(2\Delta^2+\Delta\right)n/6$。 显示英文摘要

摘要： 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 $\sum_{u\in V(G)}\rho(u)\leq \left(2\Delta^2+\Delta\right)n/6$.