A note on vertex-critical induced subgraphs of shift graphs

Abstract

Shift graphs, introduced by Erd\H{o}s and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a suprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a unique $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new and remarkably simple family of triangle-free vertex-critical graphs of arbitrarily large chromatic number.