Smaller Subgraphs of Minimum Degree $k$

Abstract

In 1990 Erdős, Faudree, Rousseau and Schelp proved that for $k \ge 2$, every graph with $n \ge k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n/6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $\epsilon_k>0$. We make progress towards their conjecture by showing that one can remove at least order of $\Omega(n/\log n)$ many vertices.