Partite Saturation of Complete Graphs

TL;DR

This paper investigates the minimum edges in k-partite graphs avoiding a complete subgraph, establishing bounds for the asymptotic behavior of the saturation function and disproving a related conjecture.

Contribution

It introduces a new function α(k,r) characterizing the asymptotic saturation number and provides tight bounds, extending previous results and resolving open questions.

Findings

Defined the function α(k,r) for saturation number asymptotics.

Established bounds for α(k,r) depending on k and r.

Proved the tightness of the lower bound for infinitely many r and all k ≥ 2r-1.

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 $K_r$-free but the addition of an edge joining any two non-adjacent vertices from different parts creates a $K_r$. Improving recent results of Ferrara, Jacobson, Pfender and Wenger, and generalizing a recent result of Roberts, we define a function $\alpha(k,r)$ such that $sat(n,k,r) = \alpha(k,r)n + o(n)$ as $n \rightarrow \infty$. Moreover, we prove that \[ k(2r-4) \le \alpha(k,r) \le \begin{cases} (k-1)(4r-k-6) &\text{ for }r \le k \le 2r-3, \\(k-1)(2r-3) &\text{ for }k \ge 2r-3, \end{cases} \] and show that the lower bound is tight for infinitely many values of $r$ and every $k\geq 2r-1$. This allows us to prove that, for these values, $sat(n,k,r) = k(2r-4)n + O(1)$ as $n \rightarrow \infty$. Along the way, we disprove a conjecture and answer a question of the first set of authors mentioned above.