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

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 ${\chi }_{\mathrm{OCF}}^{\prime }(G)$, or ${\chi }_{\mathrm{pOCF}}^{\prime }(G)$ if we additionally require $c$ to be proper. Our main result implies in particular that ${\chi }_{\mathrm{OCF}}^{\prime }(G)\le {\log }_2\Delta +O(\ln \ln \Delta )$ 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 ${\chi }_{\mathrm{pOCF}}^{\prime }(G)\le \Delta +O(\ln \Delta )$ 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.

18 pages