Bollobás-Erdős-Tuza conjecture for graphs with no induced $K_{s,t}$

Abstract

A widely open conjecture proposed by Bollobás, Erdős, 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.

5 pages