Eigenvalue bounds for the quantum chromatic number of graph powers
Aida Abiad, Benjamin Jany
TL;DR
The paper defines the quantum $k$-distance chromatic number $χ_{kq}(G)$ as the quantum analogue of distance-$k$ coloring and proves that three classical eigenvalue bounds for $χ_k(G)$ extend to $χ_{kq}(G)$. Using a pinching framework and spectral polynomials, it derives a first inertial-type bound, a second inertial-type bound for $k$-partially walk-regular graphs, and a Hoffman ratio-type bound for $χ_{kq}(G)$, with optimization of the underlying polynomial $p$ via MILP/LP. Consequently, it identifies graph classes where $χ_{kq}(G)=χ_k(G)$ and demonstrates how existing classical optimization tools can be repurposed to bound the quantum parameter. The work thus advances the understanding of quantum colorings, provides practical methods to bound $χ_{kq}$, and highlights avenues for separating quantum and classical chromatic parameters in broader graph families.
Abstract
The quantum chromatic number, a generalization of the chromatic number, was first defined in relation to the non-local quantum coloring game. We generalize the former by defining the quantum $k$-distance chromatic number $χ_{kq}(G)$ of a graph $G$, which can be seen as the quantum chromatic number of the $k$-th power graph, $G^k$, and as generalization of the classical $k$-distance chromatic number $χ_k(G)$ of a graph. It can easily be shown that $χ_{kq}(G) \leq χ_k(G)$. In this paper, we strengthen three classical eigenvalue bounds for the $k$-distance chromatic number by showing they also hold for the quantum counterpart of this parameter. This shows that several bounds by Elphick et al. [J. Combinatorial Theory Ser. A 168, 2019, Electron. J. Comb. 27(4), 2020] hold in the more general setting of distance-$k$ colorings. As a consequence we obtain several graph classes for which $χ_{kq}(G)=χ_{k}(G)$, thus increasing the number of graphs for which the quantum parameter is known.