Single‐conflict colouring

Given a multigraph, suppose that each vertex is given a local assignment of [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] is [Formula: see text] as [Formula: see text]. This is sharp up to the logarithmic factor.