Necessary Spectral Conditions for Coloring Hypergraphs
J. Combinatorial Computing and Machine Computing, 88 (2014), pp. 73-84 Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
11.12.2014
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Subjects | |
Online Access | Get full text |
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Summary: | J. Combinatorial Computing and Machine Computing, 88 (2014), pp.
73-84 Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$
obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and
$\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of
$G$ respectively. Lov\'asz later showed that $\chi(G) \le 1 -
\frac{\lambda_1}{\lambda_{n}}$ for any (perhaps negatively) weighted adjacency
matrix.
In this paper, we give a probabilistic proof of Lov\'asz's theorem, then
extend the technique to derive generalizations of Hoffman's theorem when
allowed a certain proportion of edge-conflicts. Using this result, we show that
if a 3-uniform hypergraph is 2-colorable, then $\bar d \le
-\frac{3}{2}\lambda_{\min}$ where $\bar d$ is the average degree and
$\lambda_{\min}$ is the minimal eigenvalue of the underlying graph. We
generalize this further for $k$-uniform hypergraphs, for the cases $k=4$ and
$5$, by considering several variants of the underlying graph. |
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DOI: | 10.48550/arxiv.1412.3855 |