A note on vertex-critical induced subgraphs of shift graphs

TL;DR

This paper characterizes the unique minimal vertex-critical subgraphs within shift graphs, revealing a simple family of triangle-free graphs with arbitrarily large chromatic number and a distinctive critical structure.

Contribution

It explicitly describes and proves the uniqueness of the smallest k-vertex-critical subgraph in shift graphs, a novel insight into their structure.

Findings

Identifies a unique k-vertex-critical subgraph in each shift graph for all k ≥ 1

Provides an explicit description of these critical subgraphs

Establishes the critical subgraphs as a new family of triangle-free graphs with large chromatic number

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.