From light edges to strong edge-colouring of 1-planar graphs
Résumé
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 Erdős and Nešetřil, stated back in the 80's, asserts that every graph with maximum degree ∆ should have strong chromatic index at most roughly 1.25∆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∆, 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 ∆ and strong chromatic index roughly 6∆. As an upper bound, we prove that the strong chromatic index of a 1-planar graph with maximum degree ∆ is at most roughly 24∆ (thus linear in ∆). In the course of proving the latter result, we prove, towards a conjecture of Hudák and Šugerek, that 1-planar graphs with minimum degree 3 have edges both of whose ends have degree at most 29.
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