Topology of acyclic complexes of tournaments and coloring
We prove that the acyclic complex \(\mathrm{Acy }(T)\) 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 \(\mathrm{Acy }(T)\), then the (acyclic) chromatic number of \(T\) satisfies \(\chi (T)\leqslant c(d+1)^{\frac{1}{\epsilon }-1}-1\) for some \(1.62<c\leqslant 2\), and by way of an example, we verify that the provided upper bound is tight.
acyclic complex tournament acyclic coloring 13f55 05c15 05c20