The Gyori-Lovasz theorem

TL;DR

The paper presents a clear exposition of Gyori's proof of a theorem ensuring the existence of disjoint connected subgraphs with specified sizes and vertices in a k-connected graph.

Contribution

It provides a self-contained, detailed explanation of Gyori's theorem and its proof, which was previously only available in original form.

Findings

Theorem guarantees disjoint connected subgraphs with specified properties.

Proof is simplified and made accessible for broader understanding.

The result applies to k-connected graphs with arbitrary vertex and size constraints.

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.