### Abstract

Given a simple bipartite graph G and an integer t≥2, we derive a formula for the maximum number of edges in a subgraph H of G so that H contains no node of degree larger than t and H contains no complete bipartite graph K_{t,t} as a subgraph. In the special case t=2 this fomula was proved earlier by Király (Square-free 2-matching in bipartite graphs, Technical Report of Egerváry Research Group, TR-2001013, November 1999 (www.cs.elte.hu/egres)), sharpening a result of Hartvigsen (in: G. Cornuejols, R. Burkard, G.J. Woeginger (Eds.), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 1610, Springer, Berlin, 1999, pp. 234-240). For any integer t≥2, we also determine the maximum number of edges in a subgraph of G that contains no complete bipartite graph, as a subgraph, with more than t nodes. The proofs are based on a general min-max result of Frank and Jordán (J. Combin. Theory Ser. B 65(1) (1995) 73) concerning crossing bi-supermodular functions.

Original language | English |
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Pages (from-to) | 337-346 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 131 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 12 2003 |

Event | Submodularity - Atlanta, GA, United States Duration: Aug 1 2000 → Aug 1 2000 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics