[Paper Review] $K_{s,t}$-saturated bipartite graphs

Wenying Gan, Dániel Korándi+1 more|arXiv (Cornell University)|Feb 11, 2014

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper establishes an asymptotically tight lower bound on the minimum number of edges in a $K_{s,t}$-saturated bipartite graph, proving it is at least $(s+t-2)n - (s+t-2)^2$. The authors resolve the first open case of the Moshkovitz–Shapira conjecture for $K_{2,3}$-saturation and confirm the conjecture up to an additive constant, using structural analysis of neighborhoods and core subgraphs in extremal constructions.

ABSTRACT

An $n$-by-$n$ bipartite graph is $H$-saturated if the addition of any missing edge between its two parts creates a new copy of $H$. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a $K_{s,s}$-saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a $K_{(s,t)}$-saturated bipartite graph is $n^2-(n-s+1)(n-t+1)$, where $K_{(s,t)}$ is the "ordered" complete bipartite graph with $s$ vertices in the first color class and $t$ vertices in the second. However, the very natural question of determining the minimum number of edges in the unordered $K_{s,t}$-saturated case remained unsolved. This problem was considered recently by Moshkovitz and Shapira who also conjectured what its answer should be. In this short paper we give an asymptotically tight bound on the minimum number of edges in a $K_{s,t}$-saturated bipartite graph, which is only smaller by an additive constant than the conjecture of Moshkovitz and Shapira. We also prove their conjecture for $K_{2,3}$-saturation, which was the first open case.

Motivation & Objective

• To determine the minimum number of edges in an unordered $K_{s,t}$-saturated bipartite graph, a problem left open after prior work on ordered saturation.
• To resolve the first open case of the Moshkovitz–Shapira conjecture for $K_{2,3}$-saturation.
• To provide an asymptotically tight lower bound that is within an additive constant of the conjectured extremal value.
• To analyze the structural constraints of $K_{s,t}$-saturated graphs using neighborhood and core subgraph arguments.

Proposed method

• The authors use a structural decomposition of $K_{s,t}$-saturated graphs by defining a 'core' subgraph and analyzing neighborhood constraints of key vertices.
• They apply case analysis based on whether the addition of a missing edge creates a $K_{(s,t)}$ or $K_{(t,s)}$ subgraph, depending on vertex class assignments.
• A contradiction argument is used by assuming a minimal counterexample and deriving edge count violations via neighborhood overlaps.
• The proof leverages a lemma that bounds the number of edges in a core subgraph with specific degree and connectivity properties.
• The method involves identifying vertices with limited external neighbors and using these to force structural constraints across the graph.
• The analysis is extended to the $K_{2,3}$ case by constructing a detailed neighborhood configuration and showing that any deviation leads to contradiction.

Experimental results

Research questions

• RQ1 What is the minimum number of edges in an unordered $K_{s,t}$-saturated bipartite graph for fixed $s \leq t$?
• RQ2 Does the Moshkovitz–Shapira conjecture that the saturation number is $ (s+t-2)n - \lfloor ((s+t-2)/2)^2 \rfloor $ hold up to an additive constant?
• RQ3 Is the conjectured lower bound tight for the $K_{2,3}$-saturation case?
• RQ4 Can structural constraints in $K_{s,t}$-saturated graphs be used to derive asymptotically tight edge bounds?
• RQ5 What is the relationship between ordered and unordered bipartite saturation numbers?

Key findings

• The paper proves that the minimum number of edges in a $K_{s,t}$-saturated $n \times n$ bipartite graph is at least $ (s+t-2)n - (s+t-2)^2 $, which is within an additive constant of the Moshkovitz–Shapira conjecture.
• The conjecture is confirmed in the first open case: $K_{2,3}$-saturated graphs require at least $3n - 8$ edges.
• The proof shows that any $K_{s,t}$-saturated graph must contain a core subgraph with at least $s+t-2$ vertices and specific neighborhood constraints.
• The authors demonstrate that the ordered and unordered saturation numbers differ only by an additive constant, supporting the Moshkovitz–Shapira conjecture.
• The structural analysis reveals that vertices in the core must have limited external neighbors, leading to a contradiction if the edge count falls below the derived bound.
• The result implies that extremal $K_{s,t}$-saturated graphs are highly constrained in their neighborhood configurations, limiting possible constructions.