episciences.org_5404_1653752834 1653752834 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Discrete Mathematics & Theoretical Computer Science 1365-8050 01 09 2020 vol. 22 no. 1 Graph Theory From light edges to strong edge-colouring of 1-planar graphs Julien Bensmail François Dross Hervé Hocquard Eric Sopena A strong edge-colouring of an undirected graph \$G\$ is an edge-colouring where every two edges at distance at most~\$2\$ receive distinct colours. The strong chromatic index of \$G\$ is the least number of colours in a strong edge-colouring of \$G\$. A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the \$80\$'s, asserts that every graph with maximum degree \$\Delta\$ should have strong chromatic index at most roughly \$1.25 \Delta^2\$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly \$4\Delta\$, and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of \$1\$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of \$1\$-planar graphs with maximum degree~\$\Delta\$ and strong chromatic index roughly \$6\Delta\$. As an upper bound, we prove that the strong chromatic index of a \$1\$-planar graph with maximum degree \$\Delta\$ is at most roughly \$24\Delta\$ (thus linear in \$\Delta\$). The proof of this result is based on the existence of light edges in \$1\$-planar graphs with minimum degree at least~\$3\$. 01 09 2020 5404 https://hal.archives-ouvertes.fr/hal-02112188v3 https://hal.archives-ouvertes.fr/hal-02112188v2 https://hal.archives-ouvertes.fr/hal-02112188v1 10.23638/DMTCS-22-1-2 https://dmtcs.episciences.org/5404 https://dmtcs.episciences.org/5985/pdf