Some problems on induced subgraphs

TL;DR

This paper explores various open problems in the theory of induced subgraphs, including bounds on chromatic number, perfect graph partitions, and conjectures related to graph coloring and structure.

Contribution

It presents a survey of several unresolved problems in induced subgraph theory, highlighting recent conjectures and potential research directions.

Findings

Identifies bounds for chromatic number based on clique number in specific graph classes.

Introduces the concept of perfect chromatic number and discusses its properties.

Examines conjectures related to Erdős-Hajnal and Gyárfás on graph coloring and structure.

Abstract

We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length $3$ or $4$. The second problem is about the perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect it it induces a perfect graph.) The third problem is on antichains in the induced subgraph ordering. The fourth problem is on graphs in which the difference between the chromatic number and the clique number is at most one for every induced subgraph of the graph. The fifth problem is on a weakening of the notorious Erd\H{o}s-Hajnal conjecture. The last problem is on a conjecture of Gy\'{a}rf\'{a}s about $\chi$-boundedness of a particular class of graphs.