On a bottleneck bipartition conjecture of Erdős

Abstract

For a graphG, let γ(U,V)=max{e(U), e(V)} for a bipartition (U, V) ofV(G) withUυV=V(G),UφV=Ø. Define γ(G)=min_(U,V ){γ(U,V)}. Paul Erdős conjectures$\gamma (G)/e(G) \leqslant {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} + O\left( {1/\sqrt {e(G)} } \right)$. This paper verifies the conjecture and shows$\gamma (G)/e(G) \leqslant {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}\left( {1 + \sqrt {2/e(G)} } \right)$.