On the proper rainbow saturation numbers of cliques, paths, and odd cycles

TL;DR

This paper investigates the minimum edges needed in properly rainbow-colored graphs that avoid rainbow copies of certain subgraphs, determining exact or asymptotic values for paths, cliques, and cycles.

Contribution

It provides exact and asymptotic bounds for the proper rainbow saturation number for paths, $K_4$, larger cliques, trees with diameter at least 4, and odd cycles.

Findings

Determined $ ext{sat}^*(n,H)$ for paths up to an additive constant.

Asymptotically determined $ ext{sat}^*(n,K_4)$.

Bounded $ ext{sat}^*(n,H)$ for larger cliques, trees, and odd cycles.

Abstract

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow saturation number, denoted $\text{sat}^*(n,H)$, is the minimum number of edges in an $n$-vertex rainbow $H$-saturated graph. We determine the proper rainbow saturation number for paths up to an additive constant and asymptotically determine $\text{sat}^*(n,K_4)$. In addition, we bound $\text{sat}^*(n,H)$ when $H$ is a larger clique, tree of diameter at least 4, or odd cycle.