On rainbow saturated graphs with minimum number of edges

Abstract: Let $F$ be a fixed graph without isolated vertices. An edge-colored graph is $F$-rainbow saturated if it contains no rainbow copy of $F$, but the addition of any missing edge in any color creates a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in such a graph on $n$ vertices. We prove a dichotomy governed by isolated edges: if $F$ contains an isolated edge, then $rsat(n,F)=O(1)$ for all sufficiently large $n$, while if $F$ has no isolated edge, then $rsat(n,F)=Θ(n)$. The linear lower bound is expressed in terms of a directed weight parameter $η(F)$ and establishes the linear half of the dichotomy; in several cases it also strengthens the Cameron--Puleo type coefficient. For the bounded half, we construct rainbow saturated graphs for targets of the form $H\cup K_2$. As an application of these constructions, we determine the asymptotically tight behavior for the rainbow saturation number of the generalized friendship graph $F_{t,p,q}=tK_p\vee K_q$, proving that $rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$ for fixed $t\geq 2$, $p\geq 2$ and $q\geq 1$ as $n\to\infty$.