Proper rainbow saturation for trees

TL;DR

This paper introduces and studies the proper rainbow saturation number for trees, establishing exact and asymptotic results and revealing connections to classical saturation concepts.

Contribution

It systematically investigates the proper rainbow saturation number for trees, providing new exact and asymptotic results and linking to classical saturation theories.

Findings

Exact values for certain trees' saturation numbers

Asymptotic behavior for infinite families of trees

Connections to classical saturation and semi-saturation numbers

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 initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers.