Bollob\'{a}s-Erd\H{o}s-Tuza conjecture for graphs with no induced $K_{s,t}$

TL;DR

This paper proves a longstanding conjecture for graphs excluding an induced complete bipartite subgraph, showing that large independent sets can be intersected by a small vertex subset, using probabilistic methods.

Contribution

It establishes the Bollobás-Erdős-Tuza conjecture for graphs with no induced $K_{s,t}$, expanding understanding of independent set structures in such graphs.

Findings

The conjecture holds for graphs without induced $K_{s,t}$.

Probabilistic methods are effectively used in the proof.

Provides new insights into the structure of large independent sets.

Abstract

A widely open conjecture proposed by Bollob\'as, Erd\H{o}s, and Tuza in the early 1990s states that for any $n$-vertex graph $G$, if the independence number $\alpha(G) = \Omega(n)$, then there is a subset $T \subseteq V(G)$ with $|T| = o(n)$ such that $T$ intersects all maximum independent sets of $G$. In this paper, we prove that this conjecture holds for graphs that do not contain an induced $K_{s,t}$ for fixed $t \ge s$. Our proof leverages the probabilistic method at an appropriate juncture.