## Abstract

For integers k, n with 1 ≤ k ≤ n/2, let f(k, n) be the smallest integer t such that every t-connected n-vertex graph has a spanning bipartite k-connected subgraph. A conjecture of Thomassen asserts that f(k, n) is upper bounded by some function of k. The best upper bound for f(k, n) is by Delcourt and Ferber who proved that f(k, n) ≤ 10^{10} k^{3} log n. Here it is proved that f(k, n) ≤ 22k^{2} log n. For larger k, stronger bounds hold. In the linear regime, it is proved that for any 0 < c < ½ and all sufficiently large n, if k = ⌊cn⌋, then f(k, n) ≤ 30√cn ≤ 30√n(k + 1). In the polynomial regime, it is proved that for any1/3 ≤ α < 1 and all sufficiently large n, if k = ⌊n^{α} ⌋, then f(k, n) ≤ 9n^{(1+α)/2} ≤ 9√n(k + 1).

Original language | English |
---|---|

Article number | P1.67 |

Journal | Electronic Journal of Combinatorics |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© The author.

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics