5 research outputs found
Strong edge-colouring of sparse planar graphs
A strong edge-colouring of a graph is a proper edge-colouring where each
colour class induces a matching. It is known that every planar graph with
maximum degree has a strong edge-colouring with at most
colours. We show that colours suffice if the graph has girth 6, and
colours suffice if or the girth is at least 5. In the
last part of the paper, we raise some questions related to a long-standing
conjecture of Vizing on proper edge-colouring of planar graphs
From light edges to strong edge-colouring of 1-planar graphs
International audienceA strong edge-colouring of an undirected graph is an edge-colouring where every two edges at distance at most~ receive distinct colours. The strong chromatic index of is the least number of colours in a strong edge-colouring of . A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the 's, asserts that every graph with maximum degree should have strong chromatic index at most roughly . 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 , and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of -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 -planar graphs with maximum degree~ and strong chromatic index roughly . As an upper bound, we prove that the strong chromatic index of a -planar graph with maximum degree is at most roughly (thus linear in ). The proof of this result is based on the existence of light edges in -planar graphs with minimum degree at least~