Work

A Note on Vertex-Critical Induced Subgraphs of Shift Graphs

Year: 2026

Type: article

Abstract: Shift graphs, introduced by Erdős 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 surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a unique induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

Source: The Electronic Journal of Combinatorics

DOI: 10.37236/15178

Authors: Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski, Bjarne Toft