On Chromatic Number and Minimum Cut

TL;DR

This paper establishes a deep connection between the chromatic number of certain tree graphs derived from dense graphs and the graph's cut numbers, revealing new insights into graph coloring and partitioning.

Contribution

It proves that for large dense graphs, the chromatic number of the tree graph ${\

Findings

Chromatic number of ${\cal T}_{G,t}$ equals the (n-t+1)th cut number for large dense graphs.

Chromatic number of the spanning tree graph ${\cal T}_{G,n}$ equals the minimum cut size.

Introduces a novel proof method based on an alternating Turán number inspired by Tucker's lemma.

Abstract

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prove that if $n$ is large enough and $G$ is a dense graph, then the chromatic number of the spanning tree graph ${\cal T}_{G,n}$ is equal to the size of the minimum cut of $G$. The proof method is based on alternating Tur\'an number inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem.