On Edge-Colored Saturation Problems

Let C be a family of edge-colored graphs. A t-edge colored graph G is (C, t)saturated if G does not contain any graph in C but the addition of any edge in any color in [t] creates a copy of some graph in C. Similarly to classical saturation functions, define satt(n, C) to be the minimum number of edges in a (C, t) saturated graph. Let Cr(H) be the family consisting of every edge-colored copy of H which uses exactly r colors. In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for satt(n, Cr(Kk)) for all r, showing a sharp change in behavior when r ≥ (k−1 2 ) + 2. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine satt(n, C2(K3)) exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.