Spanning $k$-trees and the colorful Carathéodory theorem

Mikhail Bludov; Alexander Polyanskii

Spanning $k$-trees and the colorful Carathéodory theorem

Combinatorics 2026-07-01 v1

Authors: Mikhail Bludov , Alexander Polyanskii

Abstract

Very recently, using Meshulam's lemma, Blagojevi\'c proved a constrained version of the colorful Carath\'eodory theorem for joins of bipartite spanning trees and wedge of spheres. Our main contribution extends his result from joins of bipartite spanning trees with wedges of spheres to joins of spanning $k$ -trees with wedges of spheres. Our proof is elementary and avoids the topological machinery. We also discuss a homological variation of spanning $k$ -trees and some Carath\'eodory-type results for them.

Keywords

spanning k-trees colorful Carathéodory theorem Meshulam's lemma joins homological variation

Cite

@article{arxiv.2607.01143,
  title  = {Spanning \(k\)-trees and the colorful Carathéodory theorem},
  author = {Mikhail Bludov and Alexander Polyanskii},
  journal= {arXiv preprint arXiv:2607.01143},
  year   = {2026}
}