Berge-Sauer conjecture on regular subgraphs

A subgraph $H\leq G$ of a graph $G$ is a graph $H=(V,E)$ such that $V\subseteq V(G)$ and $E\subseteq E(G)$. A graph is $k$-regular if every vertex has degree $k$. Berge-Sauer conjecture. Every $4$-regular simple graph contains a $3$-regular graph. This conjecture is still open; a slight modification was established by Alon in 1984.