Rainbow Saturation for Complete Graphs
Debsoumya Chakraborti, Kevin Hendrey, Ben Lund, Casey Tompkins · SIAM Journal on Discrete Mathematics · 2024
Abstract. We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph [Formula: see text] is called [Formula: see text]- rainbow saturated if [Formula: see text] does not contain a rainbow copy of [Formula: see text] and adding an edge of any color to [Formula: see text] creates a rainbow copy of [Formula: see text]. The rainbow saturation number [Formula: see text] is the minimum number of edges in an [Formula: see text]-vertex [Formula: see text]-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that [Formula: see text] for fixed [Formula: see text]. Disproving this conjecture, we establish that for every [Formula: see text], there exists a constant [Formula: see text] such that [Formula: see text] and [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text] resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored [Formula: see text]-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are [Formula: see text]-saturated with respect to the operation of deleting one edge and adding two edges.