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

Abstract

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe ($\delta \left(G\right)\ge {10}^{{10}^{10}}$), 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 length³ 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.