Partite Saturation of Complete Graphs

António Girão, Teeradej Kittipassorn, Kamil Popielarz

Published 2017-08-04Version 1

We study the problem of determining the minimum number of edges $sat(n,k,r)$ in a $k$-partite graph with $k$ parts, each of size $n$, such that it is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a $K_r$. Improving on recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we prove that $sat(n,k,r) = \alpha(k,r)n + o(n)$ where an explicit description of $\alpha(k,r)$ is given. Moreover, we give the bounds \[ k(2r-4) \le \alpha(k,r) \le \begin{cases} (k-1)(4r-k-6) &\text{ for }r \le k \le 2r-3, \\(k-1)(2r-3) &\text{ for }k \ge 2r-3. \end{cases} \] and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. This allows us to determine $sat(n,k,r) = k(2r-4)n + C_{k,r}$ up to an additive constant for those values of $k$ and $r$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.