Weak rainbow saturation numbers of graphs

For a fixed graph H, we say that an edge-colored graph G is weakly H-rainbow saturated if there exists an ordering e_1, e_2, …, e_m of E(G) such that, for any list c_1, c_2, …, c_m of pairwise distinct colors from ℕ, the non-edges e_i in color c_i can be added to G, one at a time, so that every added edge creates a new rainbow copy of H. The weak rainbow saturation number of H, denoted by rwsat(n,H), is the minimum number of edges in a weakly H-rainbow saturated graph on n vertices. In this paper, we show that for any non-empty graph H, the limit lim_n→∞rwsat(n, H)/n exists. This answers a question of Behague, Johnston, Letzter, Morrison and Ogden [SIAM J. Discrete Math. (2023)]. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if H contains no pendant edges.