Polynomial $$\chi $$ χ -Binding Functions and Forbidden Induced Subgraphs: A Survey

Abstract

A graph G with clique number $$\omega (G)$$ and chromatic number $$\chi (G)$$ is perfect if $$\chi (H)=\omega (H)$$ for every induced subgraph H of G. A family $${\mathcal {G}}$$ of graphs is called $$\chi $$ -bounded with binding function f if $$\chi (G') \le f(\omega (G'))$$ holds whenever $$G \in {\mathcal {G}}$$ and $$G'$$ is an induced subgraph of G. In this paper we will present a survey on polynomial $$\chi $$ -binding functions. Especially we will address perfect graphs, hereditary graphs satisfying the Vizing bound ( $$\chi \le \omega +1$$ ), graphs having linear $$\chi $$ -binding functions and graphs having non-linear polynomial $$\chi $$ -binding functions. Thereby we also survey polynomial $$\chi $$ -binding functions for several graph classes defined in terms of forbidden induced subgraphs, among them $$2K_2$$ -free graphs, $$P_k$$ -free graphs, claw-free graphs, and $${ diamond}$$ -free graphs. ( [])