Highly Connected Graphs Have Highly Connected Spanning Bipartite Subgraphs

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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) ≤ 1010 k3 log n. Here it is proved that f(k, n) ≤ 22k2 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 = ⌊nα ⌋, then f(k, n) ≤ 9n(1+α)/2 ≤ 9√n(k + 1).

Original languageEnglish
Article numberP1.67
JournalElectronic Journal of Combinatorics
Volume31
Issue number1
DOIs
StatePublished - 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

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