Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounded diameter monochromatic component covers

Alexey Pokrovskiy

TL;DR

The paper studies Ryser's Conjecture for $r$-edge-coloured graphs and the stronger bounded-diameter variant proposed by Milićević. It develops a framework connecting hypergraph covers and coloured-graph components via dualities, metric-space embeddings, and the notion of Ryser-stable sequences, culminating in an equivalence theorem. Consequently, it deduces that Milićević's conjecture holds for $r=5$ and extends the non-complete case to the DeBiasio–Kamel–McCourt–Sheats generalization, with explicit diameter bounds on the covering components. This reduces diameter-bounded covering questions to Ryser-type conjectures and provides a method to transfer known Ryser results, yielding bounds of the form $9^{r^{2^{(r+oldsymbol\alpha)^{4r}}}}$ for the component diameters and pointing to further open problems about universal diameter constants.

Abstract

Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Milićević's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats.

Bounded diameter monochromatic component covers

TL;DR

The paper studies Ryser's Conjecture for -edge-coloured graphs and the stronger bounded-diameter variant proposed by Milićević. It develops a framework connecting hypergraph covers and coloured-graph components via dualities, metric-space embeddings, and the notion of Ryser-stable sequences, culminating in an equivalence theorem. Consequently, it deduces that Milićević's conjecture holds for and extends the non-complete case to the DeBiasio–Kamel–McCourt–Sheats generalization, with explicit diameter bounds on the covering components. This reduces diameter-bounded covering questions to Ryser-type conjectures and provides a method to transfer known Ryser results, yielding bounds of the form for the component diameters and pointing to further open problems about universal diameter constants.

Abstract

Ryser conjectured that every -edge-coloured complete graph can be covered by monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger -- that every -edge-coloured complete graph can be covered by monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for . We also obtain several new cases of a generalization of Milićević's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats.

Paper Structure

This paper contains 7 sections, 11 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof ideas and examples
  3. Coloured graphs and $r$-partite hypergraphs
  4. Ryser stability
  5. Metric spaces
  6. Approximate duals of hypergraphs
  7. Concluding remarks

Key Result

Theorem 1.5

For fixed $r, \alpha$ the following are equivalent.

Figures (3)

  • Figure 1: A proof of of Ryser's Conjecture for $r=3$. Each case represents a configuration of monochromatic components that can occur in a $3$-coloured $K_n$. Each case has $\leq 2$ special monochromatic components which should cover the $3$-coloured $K_n$ (the shaded coloured ovals).
  • Figure 2: A proof of Conjecture \ref{['Conjecture_Milicevic']} for $r=3$. Each case represents a configuration of vertices in a $3$-coloured $K_n$ with restrictions on the coloured distances between them. Each case has $\leq 2$ special monochromatic components of bounded diameter which should cover the $3$-coloured $K_n$ (the shaded coloured ovals).
  • Figure 3: An example of a sequence of 3-uniform, 3-partite hypergraphs that is $2$-Ryser-stable sequence relative to $\mathcal{M}_{3, 2}$ (the 3-uniform matching of 2 edges). The covers witnessing 2-Ryser stability are given by square vertices in each hypergraph.

Theorems & Definitions (37)

  • Conjecture 1.1: Ryser
  • Conjecture 1.2: Milićević, milicevic2019covering
  • Conjecture 1.3: Austin, austin2005contractive
  • Conjecture 1.4: DeBiasio-Kamel-McCourt-Sheats, debiasio2020generalizations
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 3.1
  • Definition 4.1
  • Definition 4.2
  • ...and 27 more