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Simultaneous edge-colourings

Simona Boyadzhiyska, Richard Lang, Allan Lo, Michael Molloy

Abstract

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree $Δ$, for small values of $k$ and for an infinite sequence of values of $k$. This asymptotically settles a conjecture of Cabello for $k=2$. Moreover, we show that $\sqrt k Δ+ o(Δ)$ colours always suffice, which tends to the optimal value as $k$ grows. We also show that $\ell Δ+ o(Δ)$ colours are enough when every edge appears in at most $\ell$ of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of Füredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings.

Simultaneous edge-colourings

Abstract

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the maximum degree , for small values of and for an infinite sequence of values of . This asymptotically settles a conjecture of Cabello for . Moreover, we show that colours always suffice, which tends to the optimal value as grows. We also show that colours are enough when every edge appears in at most of the graphs, which asymptotically confirms a conjecture of Cambie. Finally, our results extend to the list setting. We also find a close connection to a conjecture of Füredi, Kahn, and Seymour from the 1990s and an old problem about fractional matchings.

Paper Structure

This paper contains 10 sections, 18 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Aiming for the stars
  3. Bounded edge-sizes
  4. Bounded number of vertices
  5. Chromatic index and fractional matchings
  6. Bounds for fractional matchings
  7. From stars to general graphs
  8. Sparsification
  9. Lower bound constructions
  10. Concluding remarks

Key Result

Theorem 1.1

Let $k\geq 1$ be a fixed integer. Then all graphs $G_1,\dots,G_k$ of maximum degree at most $\Delta$ satisfy $\chi'(G_1,\dots,G_k) \leq \nu(k)\Delta +o(\Delta)$. Furthermore, there exist $\Delta$-regular stars $G_1,\dots, G_k$ with the same centre vertex such that $\chi'(G_1,\dots,G_k) \geq \nu(k)\D

Theorems & Definitions (36)

  • Theorem 1.1
  • Conjecture 1.2: Cambie
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Profile
  • Conjecture 2.3: Füredi, Kahn and Seymour
  • Definition 2.4
  • proof : Proof of \ref{['thm:simultaneous-colourings-bounded']}
  • Definition 2.5
  • ...and 26 more