Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on $\chi_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $\Delta \geq 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $\Delta$ lies between $2^{\frac{\Delta}{2}-1}$ and $(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $\Delta \leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$.

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具有度数约束的可推色数图

Klostermeyer 和 MacGillivray 在 2004 年引入了有向图的可推同态和可推色数 $\chi_p$。他们特别观察到，对于任何有向图 $\overrightarrow{G}$，我们有 $\chi_p(\overrightarrow{G }) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$，其中$\chi_o(\overrightarrow{G})$ 表示$\overrightarrow{G 的定向色数}$。这是 $\chi_p$ 的第一个一般边界。在后来的工作中进一步研究了这个参数。这项工作致力于满足特定度条件的有向图的可推色数。对于所有 $\Delta \geq 29$，我们首先证明最大度数为 $\Delta$ 的有向图的可推色数的最大值介于 $2^{\frac{\Delta}{2}-1}$ 和 $(\Delta-3) \ cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ 这意味着对同一族图的有向色数的改进界限。对于亚三次定向图，即当 $\Delta \leq 3$ 时，我们证明可推色数的最大值为~$6$ 或~$7$。我们还证明了最大平均度数小于~$3$的有向图的可推色数的最大值介于~$5$和~$6$之间。前~$7$ 的上限也作为周长至少~$6$ 的平面定向图的可推色数的上限。