**4. Concluding Remarks**

In this paper, we prove that Cayley graph T*n*(*S*) generated by transposition trees on {1, 2, ... , *n*} is (*n* − 2)-extendable and determine that the extendability number is *n* − 2, which enriches the results on the extendability of Cayley graphs. Here, the transposition generating graph of *S* is a tree. A natural problem is whether we can generalize transposition trees to general connected graphs which is worth of further investigation. We present a conjecture.

**Conjecture 1.** *Let S be a transposition generating set of the symmetric group* S*n. Then, the Cayley graph* Cay(S*n*, *S*) *is* (|*S*| − 1)*-extendable.*

**Author Contributions:** Methodology: Y.F. and S.X.; writing—original draft preparation:Y.F. and Y.X.; writing—review and editing: Y.F., Y.X., F.L. and S.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (No. 11571155, No. 11961067, No. 12071194).

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** Many thanks to the anonymous referees for their helpful comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
