Eigenvalue bounds for distance-edge colorings

TL;DR

This paper establishes sharp algebraic bounds for the distance-t chromatic index of graphs using eigenvalues and polynomial methods, with implications for the Erdős–Nešetřil conjecture.

Contribution

It introduces new spectral bounds for the distance-t chromatic index and explores their tightness, computational performance, and implications for longstanding conjectures.

Findings

Derived sharp lower bounds based on eigenvalues of the line graph.

Identified graph classes that attain the bounds exactly.

Provided computational results demonstrating the bounds' effectiveness.

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.