The present work aims at deriving theoretical guaranties on the behavior of some cross-validation procedures applied to the k-nearest neighbors (kNN) rule in the context of binary classification. Here we focus on the leave-p-out cross-validation (LpO) used to assess the performance of the kNN classifier. Remarkably this LpO estimator can be efficiently computed in this context using closed-form formulas derived by Celisse and Mary-Huard (2011). We describe a general strategy to derive moment and exponential concentration inequalities for the LpO estimator applied to the kNN classifier. Such results are obtained first by exploiting the connection between the LpO estimator and U-statistics, and second by making an intensive use of the generalized Efron-Stein inequality applied to the L1O estimator. One other important contribution is made by deriving new quantifications of the discrepancy between the LpO estimator and the classification error/risk of the kNN classifier. The optimality of these bounds is discussed by means of several lower bounds as well as simulation experiments.