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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:
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