implied volatility - Calendar Arbitrage in a Vol Surface

I am trying to determine the condition such that my implied vol surface doesn't have calendar arbitrage. I have done research and found that one such condition is that total variance should increase along the time axis. However, I want to find a different condition using the call option price or forwards, or something to that extent.

Furthermore, I do not want to assume proportional dividends, same forward moneyness, etc. The information I do know is option price and forward prices.

My approach is something as follows. Let X and Y be unknowns. at t=0, I would need to pay (or receive) $XC(t_1)+YC(t_2)$ where $C(T)=\exp(-rT)BS(F_T,K,T,r,\sigma)$. Note that I am assuming that we are working with the same strike $K$. Let $X=1$ for simplicty. At $t=1$, if $S_{t_1}K$, then my payoff would be $S_{t_1}-K-Y\exp(rT_1)C(t_2)$.

I'm not sure how I would continue my argument from here, though perhaps I want to use the fact that $C(t) \ge \exp(-rt)(F_t-K)$. I know I would first need to find out the quantity of $Y$ first.

Any help would be greatly appreciated. Jim