How to build a pillar: a proof of Thomassen's conjecture

TL;DR

This paper proves Thomassen's long-standing conjecture that extremely high minimum degree graphs contain a specific structured subgraph called a pillar, using an algorithmic approach in sublinear expanders.

Contribution

The paper provides the first proof of Thomassen's conjecture by constructing a pillar in graphs with very high minimum degree through algorithmic methods.

Findings

Proof of Thomassen's conjecture established.

Constructs a pillar in sublinear expanders.

Uses algorithmic approach for construction.

Abstract

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree more than the number of atoms in the universe ($\delta(G)\ge 10^{10^{10}}$), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of the same length, $s$ say, along with $s$ vertex-disjoint paths of the same length which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this paper, we give a proof of this conjecture by building a pillar (algorithmically) in sublinear expanders.