Extremal even-cycle-free subgraphs of the complete transposition graphs

Given graphs $G$ and $H$, the generalized Tur\'{a}n number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$ and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the complete transposition graph which is defined as the Cayley graph on the symmetric group ${\rm S}_n$ with respect to the set of all transpositions of ${\rm S}_n$.