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.

Additional Details

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

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