Splittings for symbolic powers of edge ideals of complete graphs

In this paper we study the $s$-th symbolic powers of the edge ideals of complete graphs. In particular, we provide a criterion for finding an Eliahou-Kervaire splitting on these ideals, and use the splitting to provide a description for the graded Betti numbers. We also discuss the symbolic powers and graded Betti numbers of edge ideals of parallelizations of finite simple graphs.