Given a finite simple graph without triangles, I am interested in conditions ensuring that there exists an orientation of the edges such that the following holds.

There exists no cycle $x_0,x_1,\dots,x_{2p},x_0$ of odd length such that for each $0 \leq i \leq p-1$, both edges $[x_{2i},x_{2i+1}]$ and $[x_{2i+1},x_{2i+2}]$ are oriented towards $x_{2i+1}$.

If there is no cycle of odd length at all, any orientation works. If the graph is $3$-colourable, one can also easily find such an orientation. Are there more general conditions on the graph to ensure the existence of such an orientation ?