Improved bounds for proper rainbow saturation

TL;DR

This paper introduces new bounds for the proper rainbow saturation number in graphs, connecting it to classical saturation concepts and analyzing various graph classes such as cliques, cycles, and bipartite graphs.

Contribution

It establishes novel upper bounds for the proper rainbow saturation number using classical saturation links and provides initial lower bounds, expanding understanding of rainbow saturation.

Findings

New upper bounds for $ ext{sat}^*(n,H)$ for various graphs

Connections between rainbow saturation and classical saturation numbers

Initial lower bounds and exploration of related directions

Abstract

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on $\text{sat}^*(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on $\text{sat}^*(n,H)$ and explore several other interesting directions.