## Abstract

A class G of graphs is said to be χ. bounded if there is a function f:N→R such that for all G∈G and all induced subgraphs H of G, χ(H). ≤. f(ω(H)). In this paper, we show that if G is a χ-bounded class, then so is the closure of G under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if G is χ-bounded by a polynomial (respectively: exponential) function, then the closure of G under substitution is also χ-bounded by some polynomial (respectively: exponential) function. In addition, we show that if G is a χ-bounded class, then the closure of G under the operations of gluing along a clique and gluing along a bounded number of vertices together is also χ-bounded, as is the closure of G under the operations of substitution and gluing along a clique together.

Original language | English (US) |
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Pages (from-to) | 567-586 |

Number of pages | 20 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 103 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2013 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Coloring
- Connectivity
- Graph operations
- Homogeneous set
- Substitution
- χ-bounded