Transversals in a collections of trees

TL;DR

This paper studies the maximum size of collections of trees or stars on a fixed vertex set that avoid rainbow subgraphs, providing exact values and characterizations of extremal configurations.

Contribution

It determines exact maximum sizes of rainbow-free collections in families of stars and trees of equal order, and characterizes all extremal cases.

Findings

Exact maximum sizes for rainbow-free collections of stars and trees.

Characterization of all extremal configurations.

Results applicable when the number of vertices divides the size of the collection.

Abstract

Let $\mathcal{S}$ be a fixed family of graphs on vertex set $V$ and $\mathcal{G}$ be a collection of elements in $\mathcal{S}$. We investigated the transversal problem of finding the maximum value of $|\mathcal{G}|$ when $\mathcal{G}$ contains no rainbow elements in $\mathcal{S}$. Specifically, we determine the exact values when $\mathcal{S}$ is a family of stars or a family of trees of the same order $n$ with $n$ dividing $|V|$. Further, all the extremal cases for $\mathcal{G}$ are characterized.