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arXiv · 2601.16342

A note on vertex-critical induced subgraphs of shift graphs

Tomáš Kaiser·Matěj Stehlík·Riste Škrekovski
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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 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.

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Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski. 2026-01-22. A note on vertex-critical induced subgraphs of shift graphs. https://arxiv.org/abs/2601.16342

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