On the anti-Ramsey numbers for spanning trees (Q2732627)

Let \({\mathcal T}_n\) denote the family of all trees with \(n\) vertices, and let \({\mathcal T}_n^*\) denote the family of forests obtained from \({\mathcal T}_n\) by deleting from each graph of \({\mathcal T}_n\) an edge in all possible ways. It is shown that \(\text{AR}(n, {\mathcal T}_n) - \text{ext}(n, {\mathcal T}_n^*) = 1\), where \(\text{AR}(n, {\mathcal T}_n)\) is the anti-Ramsey number for the family \({\mathcal T}_n\) and \(\text{ext}(n, {\mathcal T}_n^*)\) is the Turán extremal number for the family \({\mathcal T}_n^*\). This is an example that illustrates that there is a relationship between anti-Ramsey numbers and Turán extremal numbers, as indicated by Erdős.