## Abstract

We study a function on graphs, denoted by "Gamma", representing vectorially the domination number of a graph, in a way similar to that in which the Lovsz Theta function represents the independence number of a graph. This function is a lower bound on the homological connectivity of the independence complex of the graph, and hence is of value in studying matching problems by topological methods. Not much is known at present about the Gamma function, in particular, there is no known procedure for its computation for general graphs. In this article we compute the precise value of Gamma for trees and cycles, and to achieve this we prove new lower and upper bounds on Gamma, formulated in terms of known domination and algebraic parameters of the graph. We also use the Gamma function to prove a fractional version of a strengthening of Ryser's conjecture.

Original language | English |
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Pages (from-to) | 152-170 |

Number of pages | 19 |

Journal | Journal of Graph Theory |

Volume | 70 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2012 |

Externally published | Yes |

## Keywords

- graph domination
- homological connectivity
- matching in hypergraphs
- vector representation

## ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics