Topology of acyclic complexes of tournaments and coloring. (English) Zbl 1327.13075
Author’s abstract: We prove that the acyclic complex \(\text{Acy}(T)\) of any trisectionable tournament \(T\) is homotopy equivalent to a wedge of spheres, and show that there exists a fix number \(0 < \epsilon < 1\) such that if \(T\) is a trisectionable tournament and \(d\) is the highest dimension of a sphere occurring in such a decomposition for \(\text{Acy}(T)\), then the (acyclic) chromatic number of \(T\) satisfies \(\chi(T)\leq c(d + 1)^{\frac{1}{\epsilon}-1}-1\) for some \(1.62 < c \leq 2\), and by way of an example, we verify that the provided upper bound is tight.
Reviewer: S. A. Seyed Fakhari (Tehran)
MSC:
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 05C15 | Coloring of graphs and hypergraphs |
| 05C20 | Directed graphs (digraphs), tournaments |
Software:
OEISReferences:
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