Abstract | Abstract We study the vanishing ideal of the parametrized algebraic toric set associated to the complete multipartite graph G = K_α_1 , … , α_r over a finite field of order q. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in G and which generate the toric algebra of G (type II generators) and a set of binomials of degree q − 1 obtained combinatorially from G (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo–Mumford regularity is equal to max α_1 ( q − 2 ) , … , α_r ( q − 2 ) , ⌈ ( n − 1 ) ( q − 2 ) / 2 ⌉ , where n = α_1 + ⋯ + α_r . |