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

TL;DR

This paper establishes asymptotically tight bounds for open conflict-free edge colourings in nearly regular graphs, improving previous bounds and applying to random graphs, with implications for graph colouring theory.

Contribution

It provides new asymptotically optimal bounds for conflict-free and proper conflict-free edge colourings in nearly regular graphs, using decomposition and probabilistic methods.

Findings

hieved hord bounds h for h conflict-free edge colourings

Bounded h for proper conflict-free colourings in terms of maximum degree h

Results extend to random graphs, enhancing understanding of colouring complexities

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 $\chi'_{\rm OCF}(G)$, or $\chi'_{\rm pOCF}(G)$ if we additionally require $c$ to be proper. Our main result implies in particular that $\chi'_{\rm OCF}(G) \le \log_2 \Delta + O(\ln\ln\Delta)$ for nearly regular graphs $G$ with maximum degree $\Delta$, which is asymptotically optimal, as witnessed by the complete graphs. For proper colourings, we moreover show that $\chi'_{\rm pOCF}(G) \le \Delta + O(\ln \Delta)$ 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.