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On the chromatic number of a simplicial complex. (English) Zbl 1399.05227
The chromatic number $$\chi(G)$$ of a graph $$G$$ is the minimal number of colors needed to color its vertices $$V$$ in such a way that no edge is monochromatic. It has been studied extensively by different methods. The spectral method bounds $$\chi(G)$$ by means of the spectra of various operators is defined on the graph. An advantage of the method is that the spectrum of an operator on a finite graph can be calculated in polynomial time, while the problem of finding the chromatic number of a graph is NP-complete. The Laplacian $$\Delta$$ of a graph $$G$$ is an operator on the space $$C^0$$ of the real-valued functions on the vertex set $$V$$ of $$G$$ acting by the rule $\Delta f(v)=\operatorname{deg}v\cdot f(v)-\sum_{u\sim v}f(u),$ where $$\text{deg}v$$, $$v\in V$$, is the number of vertices adjacent to $$v$$, and $$u\sim v$$ means that $$u$$ and $$v$$ are connected by an edge. The paper A. J. Hoffman [in: Graph theory and its applications. Proceedings of an advanced seminar conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, October 13–15, 1969. New York - London: Academic Press. 79–91 (1970; Zbl 0221.05061)] gives a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix.
In the paper under review, the author establishes a higher dimensional version of this result and gives a lower bound on the chromatic number of a pure $$d$$-dimensional simplicial complex in terms of the spectra of the higher Laplacian operators introduced in B. Eckmann, [Comment. Math. Helv. 17, 240–255 (1945; Zbl 0061.41106)].
The finite pure $$d$$-dimensional abstract simplicial complex is a family $$X$$ of subsets (called faces) of a finite vertex set $$V$$ closed under taking subsets, such that every maximal subset in the family is of size $$d+1$$. The chromatic number $$\chi(X)$$ is the least number of colors needed to color its vertices in such a way that no maximal face is monochromatic. In the special case of a pure $$d$$-dimensional simplicial complex with a complete $$(d-1)$$-skeleton, a different shorter proof is given.

##### MSC:
 05E45 Combinatorial aspects of simplicial complexes 05A20 Combinatorial inequalities 05C15 Coloring of graphs and hypergraphs
##### Citations:
Zbl 0221.05061; Zbl 0061.41106
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##### References:
 [1] M. Deza and P. Frankl: On the maximum number of permutations with given maximal or minimal distance, Journal of Combin · Zbl 0352.05003 [2] B. Eckmann: Harmonische Funktionen und Randwertaufgaben in einem Komplex, Commentar · Zbl 0061.41106 [3] A. J. Hoffman: On Eigenvalues and Colorings of Graphs, Graph Theory and its Applications, (ed: B. Harris), Academic Press, (1970), 79–91. [4] D. Horak and J. Jost: Spectra of combinatorial Laplace operators on simplicial complexes, · Zbl 1290.05103 [5] L. LovĂˇsz: On the Shannon capacity of a graph, IEEE Transactions of Information Theory, IT-25(1), (1979), 1–7. · Zbl 0395.94021 [6] A. Lubotzky, R. Phillips and P. Sarnak: Ramanuj · Zbl 0661.05035 [7] O. Parzanchevski, R. Rosenthal, R. J. Tessler: Isoperimetric Inequalities in Simplicial · Zbl 1389.05174 [8] P. Renteln: On the Spectrum of the Derangement Graph, Electronic · Zbl 1183.05047 [9] H. S. Wilf: The Eigenvalues of a Graph and Its Chromatic Number, Journal of the Lon · Zbl 0144.45202 [10] P. Wocjan and C. Elphick: New Spectral Bounds on the Chromatic Number Encompassing all Eigenvalues of the Adjacency Matrix. Electronic · Zbl 1295.05112
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