Shift Graphs, Chromatic Number and Acyclic One-Path Orientations

TL;DR

This paper explores properties of shift graphs, establishing bounds on chromatic numbers of certain induced subgraphs and analyzing the acyclic one-path (AOP) orientation property, revealing new structural insights.

Contribution

It improves bounds on chromatic numbers of K_{a,b}-free subgraphs in shift graphs and constructs examples of graphs with high chromatic number and odd-girth that either do or do not have the AOP property.

Findings

K_{a,b}-free subgraphs of shift graphs have chromatic number O(log(a+b))

Shift graphs G_{n,2} do not have the AOP property for n ≥ 9

Existence of high chromatic, odd-girth subgraphs with AOP property

Abstract

Shift graphs, which were introduced by Erd\H{o}s and Hajnal, have been used to answer various questions in extremal graph theory. In this paper, we prove two new results using shift graphs and their induced subgraphs. 1. Recently Girao [Combinatorica2023], showed that for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$, in which every $F$-free induced subgraph has chromatic number at most $c_F$. We significantly improve the value of the constant $c_F$ for the special case where $F$ is the complete bipartite graph $K_{a,b}$. We show that any $K_{a,b}$-free induced subgraph of the triangle-free shift graph $G_{n,2}$ has chromatic number bounded by $\mathcal{O}(\log(a+b))$. 2. An undirected simple graph $G$ is said to have the AOP Property if it can be acyclically oriented such that there is at most one directed path between any two vertices. We prove that the shift graph $G_{n,2}$ does not have the AOP property for all $n\geq 9$. Despite this, we construct induced subgraphs of shift graph $G_{n,2}$ with an arbitrarily high chromatic number and odd-girth that have the AOP property. Furthermore, we construct graphs with arbitrarily high odd-girth that do not have the AOP Property and also prove the existence of graphs with girth equal to $5$ that do not have the AOP property.