Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ colors bounded above by $\beta n$ with some fixed $\beta>1$?

What if we ask to cover only every every simple even cycle with alternating permutation (http://mathworld.wolfram.com/AlternatingPermutation.html) of vertices?