Partite Saturation of Complete Graphs

Abstract

We study the problem of determining sat(n, k, r), the minimum number of edges in a k-partite graph G with n vertices in each part such that G is Kr-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a Kr. Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function α(k, r) such that sat(n, k, r) = α(k, r)n + o(n) as n → ∞. Moreover, we prove thatand show that the lower bound is tight for infinitely many values of r and every k ≥ 2r − 1. This allows us to prove that, for these values, sat(n, k, r) = k(2r − 4)n + O(1) as n → ∞. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.