The oriented relative clique number of triangle-free planar graphs is 10

Abstract: In relation to oriented coloring and chromatic number, the parameter oriented relative clique number of an oriented graph $\overrightarrow{G}$, denoted by $\omega_{ro}(\overrightarrow{G})$, is the main focus of this work. We solve an open problem mentioned in the recent survey on oriented coloring by Sopena (Discrete Mathematics 2016), and positively settle a conjecture due to Sen (PhD thesis 2014), by proving that the maximum value of $\omega_{ro}(\overrightarrow{G})$ is $10$ when $\overrightarrow{G}$ is a planar graph.