Minimizing the number of edges in $K_{s,t}$-saturated bipartite graphs

TL;DR

This paper investigates the minimum number of edges in bipartite graphs that are saturated with respect to $K_{s,t}$, confirming a conjecture for the case $s=t-1$ and improving bounds for general $s,t$.

Contribution

It confirms the conjecture for $s=t-1$ with a classification of extremal graphs and improves existing bounds for all large $n$ and general $s,t$.

Findings

Confirmed the conjecture for $s=t-1$ with extremal graph classification.

Improved bounds on the minimum edges for $K_{s,t}$-saturated bipartite graphs.

Extended results to all sufficiently large $n$ for general $s,t$.

Abstract

This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollob\'as independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered' complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered' analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Kor\'andi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Kor\'andi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$.