Eigenvalue bounds for the quantum chromatic number of graph powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916642520301568 |
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| author | Abiad, Aida Jany, Benjamin |
| author_facet | Abiad, Aida Jany, Benjamin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue bounds for the quantum chromatic number of graph powers Abiad, Aida Jany, Benjamin Combinatorics 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. |
| title | Eigenvalue bounds for the quantum chromatic number of graph powers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.02367 |