Tales of Hoffman: from a distance
Aida Abiad, Jan Meeus

TL;DR
This paper extends Hoffman’s eigenvalue bound for chromatic number to the distance-$k$ setting, introduces a polynomial optimization approach, and explores implications for quantum and vector chromatic numbers.
Contribution
It generalizes Hoffman’s bound to distance-$k$ graphs, introduces a polynomial optimization method, and connects classical and quantum coloring parameters.
Findings
Extended Hoffman bound to distance-$k$ graphs.
Proposed linear programming to optimize polynomial bounds.
Bound is sharp for certain graph classes.
Abstract
Hoffman proved that a graph with adjacency eigenvalues and chromatic number satisfies where is the smallest integer such that We extend this eigenvalue bound to the distance- setting, and also show a strengthening of it by proving that it also lower bounds the corresponding quantum distance coloring graph parameter. The new bound depends on a degree- polynomial which can be chosen freely, so one needs to make a good choice of the polynomial to obtain as strong a bound as possible. We thus propose linear programming methods to optimize it. We also investigate the implications of the new bound for the quantum distance chromatic number, showing that it is sharp for some classes of graphs. Finally, we extend the Hoffman bound to the distance…
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Taxonomy
TopicsAdvanced Graph Theory Research · Quantum Computing Algorithms and Architecture · Limits and Structures in Graph Theory
