A criterion for a graph to admit a tight substantial map into the Euclidean \(n\)-space (Q2780952)

In this paper all graphs considered are finite 1-dimensional \(CW\)-complexes. Let \(G\) be a graph and \(E^n\) the \(n\)-dimensional Euclidean space with standard metric. The continuous map \(f:G\to E^n\) is said to be polygonal if there exists a subdivision \(G'\) of \(G\) such that \(f\) maps each edge of \(G'\) homeomorphically onto a line segment of \(E^n\). A polygonal map \(f:G\to E^n\) from a connected graph \(G\) is said to be tight if the function \(uf:=u^*\circ f:G\to \mathbb{R}\), \(x\mapsto f(x)\cdot u\) has only two local extrema, a maximum and a minimum, for all \(u\in S^{n-1}\setminus \Lambda_f\), \(\Lambda_f= \{u\in S^{n-1}/p_u \circ f:G \to E^n\) is not a polygonal map\}, where \(p_u\) is the projection.NEWLINENEWLINENEWLINEIt is a fundamental problem in the theory of tight maps whether for given \(X\) and \(n \in\mathbb{N}\) there exists a tight substantial map \(f:X\to E^n\) or not.NEWLINENEWLINENEWLINEIn this note the author gives a criterion for a finite graph to admit a tight and substantial polygonal map into the \(n\)-dimensional Euclidean space. Theorem 1: Let \(G\) be a connected graph and \(n\) \((n\geq 2)\) an integer. There exists a tight substantial polygonal map \(f:G\to E^n\) if and only if \(G\) is 2-connected and contains a subgraph which is homeomorphic to \(K_{n+1}\) -- the complete graph on \(n+1\) vertices. Finally, the author gives another proof for the 3-dimensional case of theorem 1.