Eigenvalue bounds for distance-edge colorings

Abstract

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 show 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 to the Erd\H{o}s-Ne\v{s}et\v{r}il 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.