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On asymptotically tight bounds for the open conflict-free chromatic indexes of nearly regular graphs

Mateusz Kamyczura, Jakub Przybyło

TL;DR

The paper investigates open conflict-free edge colorings and their proper variants in nearly regular graphs. It develops a decomposition-based probabilistic framework combining Lovász Local Lemma, Chernoff bounds, Talagrand concentration, and matching theory to construct a partial proper edge-coloring that satisfies all edges using $\log_2\Delta + O(\ln\ln\Delta)$ colours, subsequently extended with a controlled reservoir of colours to satisfy remaining edges. This yields asymptotically tight bounds: $\chi'_{\rm OCF}(G) \le (1+o(1))\log_2\Delta$ and $\chi'_{\rm pOCF}(G) \le (1+o(1))\Delta$ for graphs with $\delta \ge \Delta - 2\sqrt{\Delta}\log_2\Delta$, with parallel results for random graphs $G(n,p)$. The methods blend graph decompositions, random partitions, and concentration tools, providing a versatile approach that also informs related conflict-free and majority-edge-colouring problems. The work further suggests conjectures and directions on improving constants and extending bounds beyond nearly-regular regimes, and discusses implications for random graphs and broader coloring models.

Abstract

An edge colouring $c$ of a graph $G$ is called conflic-free if every non-isolated edge of $G$ has a uniquely coloured neighbour in its open edge neighbourhood. The least number of colours admitting such a colouring is denoted by $χ'_{\rm OCF}(G)$, or $χ'_{\rm pOCF}(G)$ if we additionally require $c$ to be proper. Our main result implies in particular that $χ'_{\rm OCF}(G) \le \log_2 Δ+ O(\ln\lnΔ)$ for nearly regular graphs $G$ with maximum degree $Δ$, which is asymptotically optimal, as witnessed by the complete graphs. For proper colourings, we moreover show that $χ'_{\rm pOCF}(G) \le Δ+ O(\ln Δ)$ in the same regime. These results improve existing bounds stemming from related colouring models and transfer directly to random graphs' setting. The proofs combine decomposition techniques with probabilistic arguments and structural properties of edge neighbourhoods.

On asymptotically tight bounds for the open conflict-free chromatic indexes of nearly regular graphs

TL;DR

The paper investigates open conflict-free edge colorings and their proper variants in nearly regular graphs. It develops a decomposition-based probabilistic framework combining Lovász Local Lemma, Chernoff bounds, Talagrand concentration, and matching theory to construct a partial proper edge-coloring that satisfies all edges using colours, subsequently extended with a controlled reservoir of colours to satisfy remaining edges. This yields asymptotically tight bounds: and for graphs with , with parallel results for random graphs . The methods blend graph decompositions, random partitions, and concentration tools, providing a versatile approach that also informs related conflict-free and majority-edge-colouring problems. The work further suggests conjectures and directions on improving constants and extending bounds beyond nearly-regular regimes, and discusses implications for random graphs and broader coloring models.

Abstract

An edge colouring of a graph is called conflic-free if every non-isolated edge of has a uniquely coloured neighbour in its open edge neighbourhood. The least number of colours admitting such a colouring is denoted by , or if we additionally require to be proper. Our main result implies in particular that for nearly regular graphs with maximum degree , which is asymptotically optimal, as witnessed by the complete graphs. For proper colourings, we moreover show that in the same regime. These results improve existing bounds stemming from related colouring models and transfer directly to random graphs' setting. The proofs combine decomposition techniques with probabilistic arguments and structural properties of edge neighbourhoods.
Paper Structure (9 sections, 15 theorems, 66 equations)

This paper contains 9 sections, 15 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Tools
  3. Main Lemma
  4. Graph Decompositon
  5. Vertex partitions
  6. Existence of suitable matchings
  7. Main result
  8. Final results
  9. Concluding remarks

Key Result

Lemma 1

Let $\Omega$ be a finite family of events in a probability space. Assume that every event $A\in\Omega$ is mutually independent of all but at most $D$ other events in $\Omega$, and that $\Pr(A)\le p$ for each $A\in\Omega$. If then $\Pr\!\left(\bigcap_{A\in\Omega}\overline{A}\right)>0$.

Theorems & Definitions (16)

  • Lemma 1: Lovász Local Lemma
  • Lemma 2: Chernoff Bound I
  • Lemma 3: Chernoff Bound II
  • Theorem 4: Talagrand's Inequality
  • Theorem 5: Hall's marriage theorem Hall
  • Theorem 6: Berge's theorem Berge
  • Theorem 7: Przybyło--Pękała, Observation 29 in PP
  • Corollary 8
  • Lemma 9
  • Theorem 10
  • ...and 6 more