The Gyori-Lovasz theorem

Abstract

Gyori and Lovasz independently proved the following beautiful theorem. Let $k\ge2$ be an integer, let $G$ be a $k$-connected graph on $n$ vertices, let $v_1,v_2,\ldots,v_k$ be distinct vertices of $G$ and let $n_1,n_2,\ldots,n_k$ be positive integers with $n_1+n_2+\cdots+n_k=n$. Then $G$ has disjoint connected subgraphs $G_1,G_2,\ldots,G_k$ such that for $i=1,2,\ldots,k$ the graph $G_i$ has $n_i$ vertices and $v_i\in V(G_i)$. We give a self-contained exposition of Gyori's proof.