A note on vertex-critical induced subgraphs of shift graphs
Tom\'a\v{s} Kaiser, Mat\v{e}j Stehl\'ik, Riste \v{S}krekovski

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 , the smallest -chromatic shift graph contains a unique -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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Topological and Geometric Data Analysis
