For a fixed positive integer t, we consider the graph colouring problem in which edges at distance at most t are given distinct colours. We obtain sharp lower bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. Our bounds are of algebraic nature; they depend on the eigenvalues of the line graph and on a polynomial which can be found using integer linear programming methods. We provide several graph classes that attain equality for our bounds, and also present some computational results which illustrate the bound’s performance. Lastly, we investigate the implications the spectral approach has for the Erdős–Nešetřil conjecture, and derive some conditions which a graph must satisfy if we could use it to obtain a counter example through the proposed spectral methods.

We use a graph-theoretic approach which yields improvements on the known Gilbert-Varshamov (GV) bound for sum-rank-metric codes for certain parameters. In particular, we show that asymptotically Fn×mqcan be partitioned into sum-rank-metric codes whose average size is bigger than the GV bound by a logarithmic factor for these parameters. Finally, we discuss the connection of such codes to set-coloring Ramsey numbers.