χ-boundedness and related problems on graphs without long induced paths: A survey

Abstract

Given a hereditary class of graphs $G$, a function $f:N\to N$ such that $f\left(1\right)=1$ and $f\left(x\right)\ge x$, for all $x\in N$ is a $\chi$-binding function for $G$ if $\chi \left(G\right)\le f\left(\omega \left(G\right)\right)$, for each $G\in G$. The class $G$ is called $\chi$-bounded if there exists a $\chi$-binding function for $G$. The class of graphs without long induced paths is well-studied and also received a wide recognition among the researchers for the past few decades. Here we present a survey on $\chi$-boundedness for some classes of graphs without long induced paths by giving more attention to structure/decomposition theorems which led to such results, and we discuss some related well-known conjectures and other problems of interest including algorithmic complexity, and their connections to $\chi$-boundedness.