The Rainbow Saturation Number Is Linear

Given a graph H, we say that an edge-coloured graph G is H-rainbow saturated if it does not contain a rainbow copy of H, but the addition of any non-edge in any colour creates a rainbow copy of H. The rainbow saturation number rsat(n,H) is the minimum number of edges among all H-rainbow saturated edge-coloured graphs on n vertices. We prove that for any non-empty graph H, the rainbow saturation number is linear in n, thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.