Shift Graphs, Chromatic Number and Acyclic One-Path Orientations (2308.14010v3)

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.

• The paper establishes new chromatic bounds for K₍ₐ,₍b₎-free induced subgraphs of shift graphs, significantly improving previous limits.
• It presents a novel construction of induced subgraphs that exhibit the AOP property with arbitrarily high chromatic numbers and odd-girth.
• The findings offer critical insights into the interplay between acyclic orientations and graph coloring, paving the way for future research.

Exploring New Dimensions in Graph Theory: Shift Graphs and AOP Property

Introduction

Graph theory, a cornerstone of discrete mathematics, has witnessed significant advancements owing to its widespread applicability across various disciplines. This article explores recent developments in the paper of shift graphs and acyclic orientations, focusing on two pivotal results pertaining to the chromatic number and the Acyclic One-Path (AOP) property. Shift graphs, introduced by Erdős and Hajnal, have provided fascinating insights into extremal graph theory. Our discussion centers on the chromatic number of $K_{a,b}$-free induced subgraphs of shift graphs and the construction of graphs with the AOP property, highlighting the theoretical and practical implications of these findings.

Chromatic Number of $K_{a,b}$-Free Induced Subgraphs of Shift Graphs

The chromatic number, a fundamental concept in graph theory, indicates the minimum number of colors required to color a graph's vertices such that no adjacent vertices share the same color. A significant contribution of this paper is establishing bounds on the chromatic number for $K_{a,b}$-free induced subgraphs of triangle-free shift graphs $G_{n,2}$, revealing that any such subgraph possesses a chromatic number at most $a+b$. This result substantially improves the previously known upper bound of $c_F$ for the case where $F$ is a complete bipartite graph $K_{a,b}$.

Acyclic One-Path Property

The Acyclic One-Path (AOP) Property is characterized by the ability of a graph to be acyclically oriented such that there exists at most one directed path between any pair of vertices. Our exploration extends to the AOP property in the field of shift graphs. A noteworthy discovery is that the shift graph $G_{n,2}$ does not possess the AOP property for all $n \geq 9$. Furthermore, we successfully construct induced subgraphs of shift graphs that not only exhibit the AOP property but also boast arbitrarily high chromatic numbers and odd-girth, signifying a first in constructive proofs for this class of graphs.

Implications and Future Perspectives

The theoretical advancements presented herein have far-reaching implications. The improvement in the chromatic bounds of $K_{a,b}$-free induced subgraphs unveils new horizons for exploring local constraints in graph coloring problems. Meanwhile, the findings related to the AOP property provide a foundation for understanding the interplay between acyclic orientations and chromatic numbers, an area ripe for future research endeavors.

Interestingly, the paper also poses several open questions, including whether the constant $c_F$ is bounded by a function of $\chi(F)$ for complete bipartite graphs. Another intriguing aspect is the open conjecture regarding the existence of graphs with arbitrarily high girth that do not exhibit the AOP property.

Conclusion

In conclusion, this paper on shift graphs and the AOP property enriches our understanding of structural graph theory, contributing significant theoretical advancements. As we forge ahead, the exploration of open questions and the intersection with other graph parameters promises to further illuminate the intricate tapestry of graph theory.