Single-conflict colouring

Given a multigraph, suppose that each vertex is given a local assignment of $$ colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least $$ for which this is always possible given any set of local assignments we call the single-conflict chromatic number of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single-conflict chromatic number of simple graphs embeddable on a surface of Euler genus $$ is $$ as $$ . This is sharp up to the logarithmic factor.