Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on vertex-critical induced subgraphs of shift graphs

Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski

TL;DR

The paper investigates triangle-free graphs with arbitrarily large chromatic number through shift graphs $G_{N,2}$. It proves that for each $n\ge 2$, the shift graph $G_{2^n+1,2}$ contains a unique induced $(n+1)$-vertex-critical subgraph $G_{2^n+1,2}[W]$, explicitly described via interval blocks $I_\ell$ and the vertex set $W$. A combinatorial framework based on $X$-good sequences is developed to characterize $n$-colorable induced subgraphs and to identify the critical vertices. This yields a simple, infinite family of triangle-free vertex-critical graphs with unbounded chromatic number, illustrating a structural property of shift graphs that contrasts with Kneser-type constructions.

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.

A note on vertex-critical induced subgraphs of shift graphs

TL;DR

The paper investigates triangle-free graphs with arbitrarily large chromatic number through shift graphs . It proves that for each , the shift graph contains a unique induced -vertex-critical subgraph , explicitly described via interval blocks and the vertex set . A combinatorial framework based on -good sequences is developed to characterize -colorable induced subgraphs and to identify the critical vertices. This yields a simple, infinite family of triangle-free vertex-critical graphs with unbounded chromatic number, illustrating a structural property of shift graphs that contrasts with Kneser-type constructions.

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 , 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.
Paper Structure (2 sections, 4 theorems, 13 equations, 1 figure)

This paper contains 2 sections, 4 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:main']}

Key Result

Theorem 1

For every integer $n \ge 2$, the shift graph $G_ {2^n+1,2}$ contains a unique induced $(n+1)$-vertex-critical subgraph, namely $G_{2^n+1,2}[W]$.

Figures (1)

  • Figure 1: A geometric representation of the vertex set $W$ of the shift graph $G_{17,2}$. Each box with coordinates $(x,y)$ corresponds to a vertex $(x,y)$ of $G_{17,2}$. Triangular regions of the same colour correspond to pairs $(x,y)$ lying in the same interval $I_ {\ell}$; the lower-left corners of each region lie on the dotted red hyperbola.

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof