Heawood inequalities
The well-known folklore theorem states that any planar graph has a vertex of degree at most five. A generalization of this fact to other surfaces is also known (see, e.g. [\textit{A. T. White}, Graphs, groups and surfaces (Second edition), North Holland (Amsterdam 1984; Zbl 0551.05037)]) and is closely related to the Heawood inequality bounding the chromatic number of a graph embeddable in a closed surface of a given Euler characteristic. The author deals with generalizations of these inequalities to complexes of higher dimensions. Among other results he shows that a simplicial complex K can be embedded in an n-dimensional pseudomanifold \(X^ n\) (n\(\geq 2)\) only if one of its (n-2)-simplices is incident with \(\delta\) (n-1)-simplices where \(\delta <n(n+1)/n-1\) or \[ \left( \begin{matrix} \delta +n-2\\ n\end{matrix} \right)-(2/n+1)\left( \begin{matrix} \delta +n-1\\ n\end{matrix} \right)\leq \dim H_{n-1}(X^ n,Z_ 2)-1. \] For the case of surfaces these are precisely the classical bounds. The methods to establish these inequalities include the fact that Kruskal-Katona simplicial complexes maximize homology.
scientific article; zbMATH DE number 4021169
scientific article