On Edge-Colored Saturation Problems

Abstract

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C},t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $\mathcal{C}$. Similarly to classical saturation functions, define ${\mathrm{sat}}_t(n,\mathcal{C})$ to be the minimum number of edges in a $(\mathcal{C},t)$ saturated graph. Let ${\mathcal{C}}_r(H)$ 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 ${\mathrm{sat}}_t(n,{\mathcal{C}}_r(K_k))$ for all $r$, showing a sharp change in behavior when $r\ge \left(\genfrac{}{}{0ex}{}{k-1}{2}\right)+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine ${\mathrm{sat}}_t(n,{\mathcal{C}}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.