# Proper Rainbow Saturation Numbers

## Document Type

Presentation Abstract

## Presentation Date

3-11-2024

## Abstract

A graph *G *is *F-saturated *if *G* does not contain *F *as a subgraph, and is edge-maximal with regards to this property. That is, for any edge *e *that *G *is "missing", the graph *G + e *obtained by adding *e *to *G *contains one or more subgraphs isomorphic to *F*. The study of *F*-saturated graphs informs the core questions in extremal graph theory. The *extremal number ex(n, F) *is the maximum number of edges possible in an *n*-vertex graph which does not contain *F* -- that is, *ex(n, F) *is the maximum number of edges in an *n*-vertex, *F*-saturated graph. On the other hand, the *saturation number sat(n, F) *is the *minimum *number of edges in an *n*-vertex, *F*-saturated graph. Both *ex(n, F) *and *sat(n, F) *are extensively studied, and naturally generalize to a variety of settings.

In this talk, we will discuss a variation on saturation numbers which arises in an edge-colored setting. An edge coloring of a graph is an assignment of colors (typically, some subset of the positive integers) to the graph's edges. We say that an edge coloring is *proper* if any two edges which share an endpoint receive distinct colors, and is *rainbow *if any two edges receive distinct colors. In 2007, Keevash, Mubayi, Sudakov, and Verstraete introduced the *rainbow extremal number*, which combines extremal graph theory questions with edge coloring. The *rainbow extremal number *of *F *is the maximum number of edges in a graph *G *such that, under some proper edge-coloring, *G *does not contain a rainbow copy of *F*. Rainbow extremal numbers have received substantial attention over the last fifteen years, but the corresponding rainbow* saturation *question was only posed very recently. In this talk, we will introduce and motivate rainbow *F-*saturated graphs and share some new results on rainbow saturation for cycles.

## Recommended Citation

Halfpap, Anna, "Proper Rainbow Saturation Numbers" (2024). *Colloquia of the Department of Mathematical Sciences*. 677.

https://scholarworks.umt.edu/mathcolloquia/677

## Additional Details

March 11, 2024 at 3:00 p.m. Math 103