### Abstract

Solving a problem raised by Bondy and Szwarcfiter [J. Graph Theory, 72 (2013), 462-477], we prove that if the edge set of a graph G of order n can be decomposed into edge-disjoint induced copies of the path P_{4} or of the paw K_{4} - P_{3}, then the complement of G has at least cn^{3/2} edges. This lower bound is tight apart from the actual value of c, and settles the previously unsolved cases for the graphs with at most four vertices. More generally the lower bound cn^{3/2} holds for any graph without isolated vertices which is not a complete multipartite graph; but a linear upper bound is valid for any complete tripartite graph.

Original language | English |
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Pages (from-to) | 320-329 |

Number of pages | 10 |

Journal | Applicable Analysis and Discrete Mathematics |

Volume | 8 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### Keywords

- Complete multipartite graph
- Decomposition
- Extremal graph theory
- Induced subgraph

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

Halász, V., & Tuza, Z. (2014). Asymptotically optimal induced decompositions.

*Applicable Analysis and Discrete Mathematics*,*8*(2), 320-329. https://doi.org/10.2298/AADM140718009H