Improved bounds for proper rainbow saturation

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e\mathrm{\notin }E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number ${\text{sat}}^{\ast }(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on ${\text{sat}}^{\ast }(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on ${\text{sat}}^{\ast }(n,H)$ and explore several other interesting directions.