On the sphericity for the join of many graphs (Q796544)

An infinite graph can be defined on the points of the Euclidean space \(E^ n\) so that \(x(\in E^ n)\), \(y(\in E^ n)\) are adjacent exactly when \(0<| x-y|<1\). The sphericity of an abstract (finite) graph G is the smallest number n such that \(E^ n\) has a subset X which induces a subgraph \(<X>\) isomorphic to G. The join \(G_ 1+G_ 2+...+G_ m\) of some graphs means \(\overline{\bar G_ 1\cup \bar G_ 2\cup...\cup \bar G_ m}\) where \(\cup\) denotes disjoint union and \(\bar G\) is the complement of G. The main result of the paper states that the sphericity of \(G_ 1+G_ 2+...+G_ m\) does not exceed \(2n-2\) where n is the maximum of the orders of \(G_ 1\), \(G_ 2,...,G_ m\).