[Paper Review] Linearly many rainbow trees in properly edge-coloured complete graphs

Alexey Pokrovskiy, Benny Sudakov|BIROn (Birkbeck, University of London)|Mar 21, 2017

Limits and Structures in Graph Theory17 references4 citations

TL;DR

This paper proves that every properly edge-coloured complete graph $K_n$ contains at least $10^{-6}n$ edge-disjoint isomorphic rainbow spanning trees, significantly improving prior bounds on three major conjectures: Brualdi-Hollingsworth, Kaneko-Kano-Suzuki, and Constantine. The result is achieved through probabilistic and extremal combinatorial techniques, establishing a linear lower bound on the number of such trees in any proper edge-colouring.

ABSTRACT

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine's Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of $K_n$ there are $10^{-6}n$ edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly $(n-1)$-edge-coloured $K_n$ has $n/9$ edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjecture.

Motivation & Objective

• To improve the best-known quantitative bounds on the Brualdi-Hollingsworth, Kaneko-Kano-Suzuki, and Constantine conjectures regarding rainbow tree decompositions.
• To establish a linear lower bound on the number of edge-disjoint isomorphic rainbow spanning trees in any properly edge-coloured $K_n$.
• To demonstrate that such trees can be found with specified structural properties (e.g., spiders with controlled degrees), extending the scope beyond isomorphism.
• To explore the robustness of rainbow tree existence under bounded colourings, contrasting with proper colourings.

Proposed method

• A probabilistic construction method is used to embed multiple edge-disjoint rainbow trees in a properly edge-coloured $K_n$.
• The proof leverages extremal graph theory and colouring constraints to ensure edge-disjointness and rainbow properties.
• Key lemmas are applied to construct specific tree types, such as $t$-spiders, with controlled degree distributions.
• The method is adapted to handle both general proper colourings and the special case of $(n-1)$-edge-colourings of $K_n$.
• Techniques from Ramsey theory and canonical colouring are implicitly used to ensure rainbow structure.
• The argument combines structural induction with random embedding to achieve linear-sized families of trees.

Experimental results

Research questions

• RQ1 Can every properly edge-coloured $K_n$ contain $\Omega(n)$ edge-disjoint isomorphic rainbow spanning trees?
• RQ2 What is the best possible lower bound on the number of edge-disjoint rainbow spanning trees in a properly edge-coloured $K_n$?
• RQ3 Can specific tree types, such as spiders with bounded degree, be simultaneously embedded as edge-disjoint rainbow trees?
• RQ4 Does the existence of linearly many rainbow trees hold under $b$-bounded colourings, or only under proper colourings?
• RQ5 Can the results be extended to find rainbow copies of arbitrary $n$-vertex trees in $K_n$?

Key findings

• The paper establishes a lower bound of $10^{-6}n$ edge-disjoint isomorphic rainbow spanning trees in every properly edge-coloured $K_n$.
• This result simultaneously improves the best known bounds for the Brualdi-Hollingsworth, Kaneko-Kano-Suzuki, and Constantine conjectures.
• For $(n-1)$-edge-coloured $K_n$, the paper proves the existence of $n/9 - 6$ edge-disjoint rainbow trees.
• The method allows for the construction of $0.000001n$ edge-disjoint rainbow spanning spiders with degrees between $0.003n$ and $0.2n$.
• The authors show that the result does not extend to $9$-bounded colourings, as such colourings may lack even a single spanning rainbow tree of radius 2.
• A recent result by Balogh, Liu, and Montgomery confirms the existence of $\epsilon n$ edge-disjoint spanning rainbow trees, supporting the asymptotic validity of the conjectures.