Pushable chromatic number of graphs with degree constraints

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~$6$.