Rainbow Saturation for Complete Graphs

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $\Gamma$ is called $H$-rainbow saturated if $\Gamma$ does not contain a rainbow copy of $H$ and adding an edge of any color to $\Gamma$ creates a rainbow copy of $H$. The rainbow saturation number $sat(n,{R}(H))$ is the minimum number of edges in an $n$-vertex $H$-rainbow saturated graph. Gir\~{a}o, Lewis, and Popielarz conjectured that $sat(n,{R}(K_r))=2(r-2)n+O(1)$ for fixed $r\geq 3$. Disproving this conjecture, we establish that for every $r\geq 3$, there exists a constant $\alpha_r$ such that $$r + \Omega\left(r^{1/3}\right) \le \alpha_r \le r + r^{1/2} \qquad \text{and} \qquad sat(n,{R}(K_r)) = \alpha_r n + O(1).$$ Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number, and asked whether this is equal to the rainbow saturation number of $K_r$, since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant $\alpha_r$ resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Gir\~{a}o, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored $K_r$-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are $K_r$-saturated with respect to the operation of deleting one edge and adding two edges.