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

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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.

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

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Notes

  1. The essential ingredients of the proof of the result that a connected \((2K_2, { claw})\)-free graph G with independence number \(\alpha (G) \ge 3\) is perfect, are Ben Rebea’s Lemma [31] asserting for a connected \({ claw}\)-free graph with \(\alpha (G) \ge 3\), that if G contains an odd \({ antihole}\), then it contains a \({ hole}\) of length five, and the following characterisation. Under the additional assumption that G is Y-free, all graphs Y such that every connected \(({ claw},Y)\)-free graph is perfect are characterised . Namely, the following three statements are equivalent: (i) every connected \(({ claw},Y)\)-free graph G with \(\alpha (G)\ge 2\) (with \(\alpha (G)\ge 3\)) distinct from an odd cycle is perfect. (ii) Every connected \(({ claw},Y)\)-free graph G with \(\alpha (G)\ge 2\) (with \(\alpha (G)\ge 3\)) distinct from an odd cycle is \(\omega (G)\)-colourable. (iii) Y is isomorphic to \(P_4\) (to \(P_5\)) or to \({ paw}\) (to \({ hammer}\)) or to an induced subgraph of \(P_4\) or \({ paw}\) (of \(P_5\) or \({ hammer}\)) (see Fig. 2).

  2. In a subsequent section we will recall this graph family and remark that the smallest \(\chi \)-binding function \(f^*_{P_4\cup K_1}\) satisfies \(f^*_{P_4\cup K_1}(\omega )\ge \frac{R(3,\omega +1 )}{2}\). Here R(p, q) is the Ramsey number, that is the smallest integer n such that every graph G of order at least n contains an independent set with p vertices or a clique with q vertices.

  3. A graph is called perfectly divisible if every induced subgraph H of G contains a set X of vertices such that X meets all largest cliques of H and X induces a perfect graph (cf. Sect. 5). Observe that every \((P_4\cup K_1)\)-free graph is perfectly divisible.

  4. Since due to Strong Perfect Graph Theorem a graph is perfect if and only if it contains neither an odd cycle of length at least five nor its complement, we observe that \(P_4\)-free graphs is the unique graph family defined in terms of one forbidden induced subgraph being perfect, thus having \(\chi \)-binding function \(f(\omega )=\omega \). Furthermore there exists no pair of forbidden induced graphs ensuring perfectness, which generalizes \(P_4\)-free graphs. Studying triples easily yields only the saturated triple \((C_5, P_6, \overline{P_6})\), i.e. every \((C_5, P_6, \overline{P_6})\)-free graph is perfect. The sequence can easily be extended, e. g. \((C_5, C_7, P_8, \overline{P_6})\), \((C_5, P_6, \overline{C_7}, \overline{P_8})\),...yielding all saturated k-tuples with \(k>2\) of (connected) forbidden induced subgraphs which imply the absence of all odd \({ hole}\) or \({ antihole}\) of length at least five.

  5. Atminas, Lozin and Zamaraev [2] examined structural properties of \(({ claw, co}\)-claw)-free graphs.

  6. Chudnovsky et al. [26] gave a decomposition result concerning \((P_5, \bar{P_5})\)-free graphs.

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We thank an anonymous reviewer for several valuable comments.

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  1. Institut für Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, 09596, Freiberg, Germany

    Ingo Schiermeyer

  2. Institut für Nachrichtentechnik, Technische Hochschule Köln, 50678, Cologne, Germany

    Bert Randerath

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Schiermeyer, I., Randerath, B. Polynomial \(\chi \)-Binding Functions and Forbidden Induced Subgraphs: A Survey. Graphs and Combinatorics 35, 1–31 (2019). https://doi.org/10.1007/s00373-018-1999-0

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