: Two scale-free goodness-of-fit tests for exponentiality based on the recent characterization of exponential law by Yanev and Chakraborty are proposed. Test statistics are functionals of $U$-empirical processes. The first of these statistics is of integral type, it is similar to the classical statistics $\omega_n^1.$ The second one is a Kolmogorov type statistic. The limiting distribution and large deviations asymptotic of new statistics under null hypothesis are described. Their local Bahadur efficiency for parametric alternatives is calculated. The Kolmogorov type statistic is not asymptotically normal, therefore we evaluate the critical values by using Monte-Carlo methods. For small sample size efficiencies are compared with simulated powers of new tests. Also conditions of local asymptotic optimality of new statistics in the sense of Bahadur are discussed and examples of such special alternatives are given.