The rainbow saturation number is linear

TL;DR

This paper proves that the rainbow saturation number for any non-empty graph grows linearly with the number of vertices, confirming a conjecture, and also improves bounds for complete graphs.

Contribution

It establishes the linearity of the rainbow saturation number for all non-empty graphs and provides improved bounds for complete graphs, disproving previous conjectures.

Findings

Rainbow saturation number is linear in n for any non-empty graph H.

Improved upper bounds for rainbow saturation number of complete graphs.

Disproved previous conjectures on rainbow saturation bounds.

Abstract

Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number $\text{rsat}(n,H)$ is the minimum number of edges among all $H$-rainbow saturated edge-coloured graphs on $n$ vertices. We prove that for any non-empty graph $H$, the rainbow saturation number is linear in $n$, thus proving a conjecture of Gir\~{a}o, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Gir\~{a}o, Lewis, and Popielarz.