Distance edge-colourings and matchings

We consider a distance generalisation of the strong chromatic index and the maximum induced matching number. We study graphs of bounded maximum degree and Erd\H{o}s–R\'{e}nyi random graphs. We work in three settings. The first is that of a distance generalisation of an Erd\H{o}s–Ne\v{s}et\v{r}il problem. The second is that of an upper bound on the size of a largest distance matching in a random graph. The third is that of an upper bound on the distance chromatic index for sparse random graphs. One of our results gives a counterexample to a conjecture of Skupie\'{n}.