Graphs and Combinatorics

, Volume 35, Issue 1, pp 1–31 | Cite as

Polynomial \(\chi \)-Binding Functions and Forbidden Induced Subgraphs: A Survey

  • Ingo Schiermeyer
  • Bert Randerath
Original Paper


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.

Families of\(\chi \)-bound graphs are natural candidates for polynomial approximation algorithms for the vertex coloring problem. (András Gyárfás [42])


Chromatic number Perfect graphs \(\chi \)-bounded \(\chi \)-binding function Forbidden induced subgraph 



We thank an anonymous reviewer for several valuable comments.


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Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und AlgebraTechnische Universität Bergakademie FreibergFreibergGermany
  2. 2.Institut für NachrichtentechnikTechnische Hochschule KölnCologneGermany

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