On even-cycle-free subgraphs of the doubled Johnson graphs

The generalized Tur'{a}n number $\mathrm{e}\mathrm{x}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G\mathrm{.}$ It is an important extension of the classical Tur'{a}n number $\mathrm{e}\mathrm{x}(n,H)$, which is the maximum number of edges in a graph with $n$ vertices that does not contain $H$ as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs $J(n;k,k+1)$, which are bipartite subgraphs of hypercube graphs. We give an upper bound for ${\rm ex}(J(n;k,k+1),C_{2r})$withanyfixed$k\in$\mathbb{Z}$^+$andany$n\in$\mathbb{Z}$^+$with$n\geq 2k+1.$Wealsogiveanupperboundfor${\rm ex}(J(2k+1;k,k+1),C_{2r})$withany$k\in$\mathbb{Z}$^+,$where$J(2k+1;k,k+1)$isknownasdoubledOddgraph$\widetilde{O}_{k+1}.$ This bound induces that the number of edges in any $C_{2r}$-free subgraph of ${\tilde{O}}_{k+1}$ is $o(e({\tilde{O}}_{k+1}))$ for $r\ge 6,$ which also implies a Ramsey-type result.