On Edge-Colored Saturation Problems

TL;DR

This paper investigates edge-colored saturation problems, determining the order of magnitude for saturation functions in various cases, proving a conjecture, and identifying exact values and irregularities in the behavior of these functions.

Contribution

It establishes the order of magnitude for sat_t(n, \u00a0C_r(K_k)) for all r, proves a conjecture, and finds exact saturation numbers for specific cases.

Findings

Determined the order of magnitude for sat_t(n, _r(K_k)) for all r.

Proved a conjecture of Barrus et al. regarding saturation functions.

Identified irregularities in the behavior of the colored saturation function.

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 $r\geq \binom{k-1}{2}+2$. 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.