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This paper considers upper bounds on the oriented chromatic number$\chi_o(G)$, of a graph $G$ in terms of its $2$-dipath chromatic number$\chi_2(G)$, degeneracy $d(G)$, and maximum degree $\Delta(G)$. In particular,we show that for all graphs $G$ with $\chi_2(G) \leq k$ and $d(G) \leq d$,$\chi_o(G) = O(k d^2 2^d)$. This improves an upper bound of MacGillivray,Raspaud, and Swartz of the form $\chi_o(G) \leq 2^{\chi_2(G)} -1$ to apolynomial upper bound for graphs with bounded $2$-dipath chromatic number.Additionally, we give an improved asymptotic upper bound of the form $\chi_o(G)\leq 2(\ln2 + o(1))\Delta^2 2^\Delta$ for all graphs with maximum degree atmost $\Delta$, in addition to an even better bound of $\chi_o(G) \leq (2\ln 2 +1 - o(1))\Delta d 2^d$ for graphs with degeneracy at most $d$. The formerimproves results by Das, Nandi, and Sen \cite{das2017chromatic}, and Kostochka,Sopena, and Zhu \cite{kostochka1997acyclic}, while the latter improves a resultof Aravind of Subramanian \cite{aravind2009forbidden}. In both cases we improvethe prior bounds by a constant factor.

Oriented Colouring Graphs of Bounded Degree and Degeneracy