Large and judicious bisections of graphs

24 April 2012
Choongbum Lee
<p>It is very well known that every graph on $n$ vertices and $m$ edges admits a bipartition of size at least $m/2$. This bound can be improved to $m/2 + (n-1)/4$ for connected graphs, and $m/2 + n/6$ for graphs without isolated vertices, as proved by Edwards, and Erd\"os, Gy\'arf\'as, and Kohayakawa, respectively. A bisection&nbsp;of a graph is a bipartition in which the size of the two parts&nbsp;differ by at most 1. We prove that graphs with maximum degree $o(n)$ in fact admit a bisection which asymptotically achieves the above bounds.These results follow from a more general theorem,&nbsp;which can also be used to answer several questions and conjectures of Bollob\'as and Scott on&nbsp;judicious bisections of graphs.<br />Joint work with Po-Shen Loh and Benny Sudakov</p>
  • Combinatorial Theory Seminar